Combinatorics of Macdonald Polynomials

By

Martha Yip

A dissertation submitted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

(Mathematics)

at the

UNIVERSITY OF WISCONSIN – MADISON

2010

i

Abstract

This work is a study of the connection between double affine Hecke algebras and the

alcove walk model, which is used to obtain combinatorial formulas for products of Mac-

donald polynomials. Chapter 2 covers the necessary background material on root sys-

tems, Weyl groups, braid groups, Hecke algebras, and the alcove walk model. Chapter

3 uses the combinatorics of alcove walks to calculate products of monomials and in-

tertwining operators of the double affine Hecke algebra. By passing to the polynomial

representation, we obtain change of basis formulas between the nonsymmetric Macdon-

ald basis and the monomial basis for the space of multivariate Laurent polynomials.

Chapter 4 gives two product formulas for polynomials; the first expresses the product

of a nonsymmetric with a symmetric Macdonald polynomial in terms of the nonsym-

metric basis, and the second calculates the structure coefficients in the product of two

symmetric Macdonald polynomials. This latter Littlewood-Richardson rule generalizes

the results for Hall-Littlewood polynomials in terms of positively folded walks, and also

for Weyl characters in terms of the Littelmann path model. Chapter 5 concludes this

work with many examples.

ii

Acknowledgements

I am indebted to Arun Ram, who has been an inspiring teacher and a supportive mentor

to me over the past four years. I am grateful to Amos Ron, for his insight and enthusiasm

for mathematics. I would like to thank the many people from whom I have learned

so much so far, and I would like to acknowledge the Mathematics Department at the

University of Wisconsin and NSERC for their generous support.

iii

Contents

Abstract i

Acknowledgements ii

1 Introduction 1

1.1 An overview of Macdonald polynomials . . . . . . . . . . . . . . . . . . . 1

1.2 Combinatorial formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Double affine Hecke algebras 6

2.1 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Double affine Weyl groups . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 The alcove picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Double affine braid groups . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Double affine Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 The polynomial representation . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7 Alcove walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Macdonald polynomials 31

3.1 Nonsymmetric Macdonald polynomials . . . . . . . . . . . . . . . . . . . 31

3.2 Symmetric Macdonald polynomials . . . . . . . . . . . . . . . . . . . . . 39

4 Littlewood-Richardson formulas 46

iv

4.1 Littlewood-Richardson formulas . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Specialization at q = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Pieri formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Examples 61

5.1 Type A1 examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2 Type A2 examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3 Type C2 examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 Further work 89

Bibliography 91

1

Chapter 1

Introduction

1.1 An overview of Macdonald polynomials

In [26] and [29], Macdonald introduced a remarkable family of orthogonal polynomials

Pλ(q, t) associated with root systems. These polynomials are indexed by dominant

weights of the weight lattice P . For particular values of q and t, they specialize to

various well-known functions, including:

1. when q = t, they are independent of q and are Weyl characters, which are charac-

ters of irreducible highest weight representations of compact Lie groups,

2. when t = 1, they are independent of q and are orbits of the finite Weyl group in

the weight lattice (monomial symmetric functions),

3. when q = 0, up to a scalar factor, they are spherical functions (Hall-Littlewood

polynomials) for semisimple p-adic Lie groups relative to a maximal compact sub-

group (see [29]),

4. when q = tα for some α ∈ R>0 and q, t→ 1, they are symmetric Jack polynomials.

For the Type A root systems, these polynomials are a basis for the space of symmetric

polynomials, and are a common generalization of Schur polynomials, monomial sym-

metric polynomials, Hall-Littlewood polynomials, and symmetric Jack polynomials.

2

As originally defined by Macdonald, the polynomials Pλ(q, t) are characterized by

certain triangularity and orthogonality conditions with respect to the scalar product

< f, g > =1

|W0|[X0]fg∇q,t,

(see [29, Theorem 4.1] or [30, (5.3.1)] for full details), where [X0]F is the constant term

of F , and the weight ∇q,t is a certain infinite product (see [29, (5.1.28)]). A number of

conjectures were made in [29], including the scalar product conjecture, which is a formula

for < Pλ, Pλ >. Cherednik developed the theory of double affine Hecke algebras, and

used it to prove Macdonald’s conjectures in a uniform manner for all reduced affine root

systems (see [6] and [7]).

The notion of nonsymmetric Macdonald polynomials arise naturally from the study

of double affine Hecke algebras; they are the eigenfunctions of the Dunkl operators.

These polynomials were first introduced by Opdam [32] in the case q → 1 (see [30,

p.147]), and by Macdonald [28, p.202] for arbitrary q. The polynomials Eλ(q, t) are a

family of orthogonal polynomials indexed by elements of the weight lattice, and are a

basis for the space of polynomials. Cherednik [8] showed that products of intertwining

operators in the double affine Hecke algebra generate Eλ. By applying a symmetrizing

operator 10 to Eλ, one can obtain the symmetric polynomials Pλ.

Following suggestions of Cherednik, Sanderson [36] and Ion [19] gave a representation-

theoretic interpretation for the nonsymmetric Macdonald polynomials by showing that,

up to a scalar factor, Eλ(q, 0) are characters of Demazure modules of Kac-Moody affine

Lie algebras. Since Pλ(q, 0) = Eλ−(q, 0) when λ is a dominant weight (see [19, The-

orem 4.2]), Pλ(q, 0) is also a Demazure character. A different representation-theoretic

interpretation for Pλ(q, 0) was given by Garsia and Procesi [10] (see [36]).

3

1.2 Combinatorial formulas

The article [11] is an excellent survey on Macdonald polynomials, and focuses on the

combinatorial side of the theory. Classically, the Schur polynomial sλ, which is Pλ(0, 0)

of the Type A root system, can be combinatorially defined by

sλ =∑T

xwt(T )

where the sum is over all semistandard Young tableaux T of shape λ [9, p.3]. In the

representation theory of the general linear group GLn(C), sλ is the character of the

irreducible highest weight module V (λ), and the multiplicity of a weight space is given

by the number of semistandard Young tableaux with that weight.

The combinatorics of tableaux also provide formulas for the multiplicity cνλµ of the

module V (ν) in V (λ)⊗V (µ). Known as Littlewood-Richardson coefficients, the numbers

cνλµ are the structure constants for the ring of symmetric polynomials with respect to

the Schur basis:

sλsµ =∑ν

cνλµsν ,

and cνλµ is the number of Young tableaux of shape ν/λ which admits a Littlewood-

Richardson filling of type µ [9, Proposition 3, p.64].

Extending these ideas to the more general setting of complex symmetrizable Kac-

Moody algebras, Littelmann introduced the path model in [24] as a tool for calculating

their characters, and showed that it can also be used to compute Littlewood-Richardson

coefficients. Instead of a sum over tableaux, his formula for cνλµ is a sum over certain

paths in the vector space P ⊗ZR, where the endpoint (weight) of a path takes the place

of the weight of a tableau.

4

Several variations of the Littelmann path model were introduced to obtain character

formulas, including the gallery model of Gaussent-Littelmann [13], the model of Lenart-

Postnikov [23] based on λ-chains, and the alcove walk model of Ram [34] for working in

the affine Hecke algebra.

Recent advances in the combinatorial theory of Macdonald polynomials include the

work by Haglund, Haiman and Loehr ([14] and [15]) who, in the Type A case, gave

explicit combinatorial formulas for the expansion of Macdonald polynomials in terms

of monomials. These formulas are sums over fillings of tableau-like diagrams. The

paper [35] gave uniform formulas for the expansion of Macdonald polynomials of all Lie

types, and this was achieved by using the alcove walk model as a tool for expanding

products of intertwining operators of the double affine Hecke algebra.

1.3 Outline of the thesis

This work is a study of the connection between double affine Hecke algebras and the al-

cove walk model, which is used to obtain combinatorial formulas for products of Macdon-

ald polynomials. Chapter 2 covers the necessary background material on root systems,

Weyl groups, braid groups, Hecke algebras, and the alcove walk model. Chapter 3 uses

the combinatorics of alcove walks to calculate products of monomials and intertwining

operators in the double affine Hecke algebra (Theorem 3.4 and Theorem 3.6). By passing

to the polynomial representation of the double affine Hecke algebra, we obtain change

of basis formulas between the nonsymmetric Macdonald basis and the monomial basis

for the space of multivariate Laurent polynomials (Corollary 3.5 and Corollary 3.7).

The main results are proved in Chapter 4:

5

Theorem 4.1 Let Eµ be the nonsymmetric Macdonald polynomial indexed by the

weight µ, and let Pλ be the symmetric Macdonald polynomial indexed by the dominant

weight λ. Then

EµPλ =∑p

ap(q, t)E$(p),

where the sum is over alcove walks of type determined by µ and contained in the domi-

nant chamber, the coefficients ap(q, t) are rational functions in q and t, and $(p) is the

weight such that m−1$(p) is the endpoint of p.

Theorem 4.5 Let Pλ be the symmetric Macdonald polynomial indexed by the dominant

weight λ. Then

PµPλ =∑p

cp(q, t)P−w0wt(p),

where the sum is over alcove walks of type determined by µ and contained in the domi-

nant chamber, the coefficients cp(q, t) are rational functions in q and t, w0 is the longest

element of the Weyl group, and wt(p) is the weight of p.

Theorem 4.1 is a generalization of the result [16, Theorem 6.1] by Haglund, Luoto,

Mason and van Willigenburg on quasisymmetric functions, and also of the results [1,

Proposition 8, 10] by Baratta on nonsymmetric Macdonald polynomials.

Theorem 4.5 is a generalization of Schwer’s formula for spherical functions [37, The-

orem 1.3], and also of Littelmann’s formula for Weyl characters [24, Section 6].

Chapter 5 concludes this work with many examples. A number of calculations can

be made completely explicit in the case of the rank one root system of Type A1, which

corresponds to the representation theory of sl2C. Some rank two examples are also

included.

6

Chapter 2

Double affine Hecke algebras

The main references for the background material covered in this chapter are Bourbaki [3],

Kane [20], and Macdonald [30].

2.1 Root systems

An n by n matrix C with entries in Z is a Cartan matrix if:

1. ci,i = 2 for i = 1, . . . , n,

2. ci,j ≤ 0 for i 6= j,

3. there exists an invertible diagonal matrix D and a positive definite symmetric

matrix S such that C = DS.

Let h∗Z be a lattice (free Z-module) of finite rank and let hZ = Hom(h∗Z,Z) be the

dual lattice. Let the pairing 〈·, ·〉 : h∗Z × hZ → Z be defined by

〈x, y〉 = y(x), for x ∈ h∗Z, y ∈ hZ.

Let {α1, . . . , αn} ⊆ h∗Z and {α∨1 , . . . , α∨n} ⊆ hZ be finite sets such that [〈αi, α∨j 〉] is a

Cartan matrix.

7

Let h∗R = h∗Z ⊗ R and hR = hZ ⊗ R. For α ∈ h∗Z and α∨ ∈ hZ such that 〈α, α∨〉 = 2,

the map

sα : h∗Z −→ h∗Z

x 7→ x− 〈x, α∨〉α

is a reflection in the hyperplane

Hα∨ = {x ∈ h∗R | 〈x, α∨〉 = 0},

sending α to −α. The simple reflections are

si = sαi , for i = 1, . . . , n.

The Weyl group W0 is generated by s1, . . . , sn, subject to the relations

s2i = 1, and sisjsi · · · = sjsisj · · · (mij factors each side),

where π/mij is the angle between Hα∨i and Hα∨j . See [20, p.69].

Given α ∈ h∗Z, the map

sα∨ : hZ −→ hZ

y 7→ y − 〈α, y〉α∨

is adjoint to sα with respect to the pairing 〈·, ·〉, so the group generated by {sα∨1 , . . . , sα∨n}

is isomorphic to W0.

The roots R and coroots R∨ are respectively

R = W0{α1, . . . , αn}, and R∨ = W0{α∨1 , . . . , α∨n}. (2.1)

8

Assume throughout this thesis that if α ∈ R, then 2α /∈ R. A root system R

satisfying this condition is reduced. Also assume throughout that there is no nonempty

subset of R that is invariant under W0. Such a root system is irreducible.

Every root α ∈ R is either a nonnegative or a nonpositive integer combination of

the simple roots {α1, . . . , αn} (see [20, Proposition 3-4]), so R = R+ t−(R+), where the

positive roots are

R+ = {α ∈ R | α is a nonnegative linear combination of α1, . . . , αn}, (2.2)

and the negative roots are R− = −R+. The reflections in the Weyl group W0 are indexed

by R+ (see [20, Proposition 9-5]).

The fundamental weights {ω1, . . . , ωn} and fundamental coweights {ω∨1 , . . . , ω∨n} are

defined by

〈αi, ω∨j 〉 = δij, and 〈ωi, α∨j 〉 = δij.

The root lattice, coroot lattice, weight lattice, and coweight lattice are

Q =n∑i=1

Zαi, Q∨ =n∑i=1

Zα∨i , P =n∑i=1

Zωi, P∨ =n∑i=1

Zω∨i ,

so P = Hom(Q∨,Z) and P∨ = Hom(Q,Z). Moreover, Q ⊆ h∗Z ⊆ P and Q∨ ⊆ hZ ⊆ P∨

as lattices.

The dominant weights P+ and regular dominant weights P++ are

P+ = {µ ∈ P | 〈µ, α∨i 〉 ≥ 0 for all i = 1, . . . , n} =n∑i=1

Z≥0ωi, (2.3)

P++ = {µ ∈ P | 〈µ, α∨i 〉 ≥ 1 for all i = 1, . . . , n} =n∑i=1

Nωi. (2.4)

Similarly, define

Q+ = {µ ∈ Q | 〈µ, ω∨i 〉 ≥ 0 for all i = 1, . . . , n} =n∑i=1

Z≥0αi.

9

The dominance order on P is defined by

θ > ω if θ 6= ω and θ − ω ∈ Q+.

Each coroot system R∨ has a unique maximal coroot ϕ∨, characterized by the property

ϕ∨ ≥ α∨ for all α∨ ∈ R∨ (see [20, p. 120]). Analogous definitions hold for the roots R.

The elements in R∨ have at most two lengths (see [20, Lemma 10-4]), and the maximal

coroot ϕ∨ is necessarily long. This implies that the root ϕ ∈ R satisfiying 〈ϕ, ϕ∨〉 = 2

is maximal amongst the short roots. See for example, Section 5.3.

2.2 Double affine Weyl groups

Let e be the smallest positive integer which satisfies 〈h∗Z, hZ〉 ⊆ 1eZ. Let

X = {xµ | µ ∈ h∗Z} and Y = {yλ∨ | λ∨ ∈ hZ}

be abelian groups isomorphic to h∗Z and hZ respectively, with multiplication

xµxλ = xµ+λ, and yλ∨yµ∨

= yλ∨+µ∨ . (2.5)

The double affine Weyl group is

W ={qkxµwyλ

∨ | k ∈ 1eZ, µ ∈ h∗Z, w ∈ W0, λ

∨ ∈ hZ

},

subject to the relations (2.5) and

wxµ = xwµw, wyλ∨

= ywλ∨w, xµyλ

∨= q〈µ,λ

∨〉yλ∨xµ, q1/e ∈ Z(W ).

See [17, Corollary 4.6].

10

The extended affine Weyl groups

W = {wyλ∨ | w ∈ W0, λ∨ ∈ hZ} = W0 n Y, (2.6)

W∨ = {xµw | µ ∈ h∗Z, w ∈ W0} = X oW0, (2.7)

are subgroups of W , and W acts by conjugation on {qkxµ | k ∈ 1eZ, µ ∈ h∗Z}. Define

xµ+kδ = qkxµ and yλ∨+kd = q−kyλ

∨. (2.8)

Then the group W acts on the lattice h∗Z ⊕ 1eZδ, where for w ∈ W and ν = µ + kδ ∈

h∗Z ⊕ 1eZδ, wν is defined by

xwν = wxνw−1 in W . (2.9)

For α ∈ R and j ∈ N, the map

xjαsα : h∗Z −→ h∗Z

x 7→ sαx+ jα

should be viewed as a reflection in the affine hyperplane

H−α∨+jd = {x ∈ h∗R | 〈x, α∨〉 = j}. (2.10)

Given maximal coroot ϕ∨ ∈ R∨, maximal (short) root ϕ ∈ R, maximal root θ ∈ R, and

maximal (short) coroot θ∨ ∈ R∨, define

α0 = −ϕ+ δ, α∨0 = −ϕ∨ + d, s0 = yθ∨sθ ∈ W, and s∨0 = xϕsϕ ∈ W∨.

The element s∨0 is a reflection in the affine hyperplane Hα∨0 = H−ϕ∨+d. See Figure 2.1 for

an illustration.

11

The affine Weyl group Wa = W0 n Q∨ is generated by s0, s1, . . . , sn, subject to the

relations

s2i = 1, and sisjsi · · · = sjsisj · · · (mij factors each side),

where π/mij is the angle between Hα∨i and Hα∨j . See [20, p.123]. The simple affine roots

are α0, . . . , αn, and the simple affine coroots are α∨0 , . . . , α∨n .

The extended affine Weyl group W has an alternate presentation [20, p.132]

W = Wa o Π, (2.11)

where Π ∼= hZ/Q∨.

The dual version of the above statements for W holds for W∨ as well. That is,

W∨ = X oW0 = Π∨ nW∨a , (2.12)

Π∨ ∼= h∗Z/Q, and W∨a = Q o W0 is the group generated by s∨0 , s1, . . . , sn. For no-

tational convenience, we sometimes write s∨i = si for i = 1, . . . , n. See Section 5.3,

Equations (5.15) and (5.17) for an illustration.

2.3 The alcove picture

Denote the positive roots and coroots by R+ and R∨+. The positive affine coroots are

S∨+ ={α∨ + jd | α∨ ∈ R∨+, j ∈ Z≥0

}∪{−α∨ + jd | α∨ ∈ R∨+, j ∈ Z≥1

}.

The chambers of W0 are the connected components of h∗R\∪α∈R+ Hα∨ , and the alcoves of

W∨a are the connected components of h∗R\ ∪a∈S+ Ha∨ .

12

The fundamental chamber or dominant chamber is the region

C = {x ∈ h∗R | 0 < 〈x, α∨〉 for α ∈ R+} =n⋂i=1

{x ∈ h∗R | 0 < 〈x, α∨i 〉}, (2.13)

whose walls (the hyperplanes which have nonempty intersection with the closure of C)

are Hα∨1 , . . . , Hα∨n . The fundamental alcove is the region

A = {x ∈ h∗R | 0 < 〈x, α∨〉 < 1 for α ∈ R+} = C ∩ {x ∈ h∗R | 〈x, ϕ∨〉 < 1}, (2.14)

and its walls are the hyperplanes Hα∨0 , . . . , Hα∨n .

A group action is simply transitive if for any two distinct elements x, y in the set,

there exists a unique group element g such that g · x = y. By [20, Proposition 4-6], W0

acts simply transitively on the chambers, so there is a bijection

W0 ←→ {chambers}

w ↔ wC.

Similarly by [20, Proposition 11-5], W∨a acts simply transitively on the alcoves, so there

is a bijection

W∨a ←→ {alcoves}

w ↔ wA.

In the above correspondence, the elements of Π∨ ⊆ W∨ = Π∨oW∨a fix the fundamental

alcove A. Since |P/Q| = det[〈αi, α∨j 〉]1≤i,j≤n is finite, then Π∨ ∼= h∗Z/Q ⊆ P/Q is a finite

abelian group. The extended affine Weyl group W∨ acts freely transitively on |Π∨| copies

(sheets) of alcoves, so there is a bijection

W∨ ←→ {alcoves} × |Π∨|

w ↔ wA,

13

where elements in W∨a permute alcoves in the base sheet, and elements π∨j ∈ Π∨ send

the fundamental alcove to the copy of the fundamental alcove on the jth sheet. In the

alcove picture for the extended affine Weyl group W∨, the dominant chamber refers to

the region C× |Π∨|.

The periodic orientation is the orientation of the hyperplanes{Ha∨ | a∨ ∈ S∨+

}such

that

1. A is on the positive side of Hα∨ for α∨ ∈ R∨+,

2. Hα∨+jd and Hα∨ have parallel orientations.

Figure 2.1 is the alcove picture for the extended affine Weyl group W∨ for the sl2C

root system, showing the correspondence between the alcoves and the elements of W∨.

The periodic orientation is indicated by + and − on either side of the hyperplanes.

The two ways of indexing the alcoves correspond to the two presentations W∨ = Π∨ n

〈s∨0 , s1, . . . , sn〉 and W∨ = Qo 〈s1, . . . , sn〉.

Figure 2.1.

•

H−α∨+2dHα∨Hα∨+2d H−α∨+d H−α∨+3dHα∨+d

Sheet 1 ...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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+− +− +− +− +− +−1 s∨0 s∨0 s

∨1 s∨0 s

∨1 s∨0s∨1s∨1 s

∨0s∨1 s

∨0 s∨1

1 xαs∨1 xα x2αs∨1s∨1x−αx−αs∨1

Sheet π∨ ...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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+− +− +− +− +− +−π∨ π∨s∨1 π∨s∨1 s

∨0 π∨s∨1 s

∨0 s∨1π∨s∨0π∨s∨0 s

∨1π∨s∨0 s

∨1 s∨0

xωs∨1 xω x3ωs∨1 x3ωx−ωx−ωs∨1x−3ω

The length function

Given w ∈ W∨ with a reduced expression w = π∨j s∨i1· · · s∨ir , the set of positive coroots

L(w) ={π∨j α

∨i1, π∨j s

∨i1α∨i2 , π∨j s

∨i1s∨i2α

∨i3, . . . , π∨j s

∨i1· · · s∨ir−1

α∨ir}

(2.15)

14

index the hyperplanes that separate the fundamental alcove A and the alcove wA. See [30,

(2.2.1), (2.2.9)].

Remark 2.2. In Macdonald’s [30] notation,

L(w) = S(w−1) = S∨+ ∩ wS∨− = {wa∨ ∈ S∨+ | a∨ ∈ S∨−} = {b∨ ∈ S∨+ | w−1b∨ ∈ S∨−}.

(2.16)

The length of w is

`(w) = |L(w)|, (2.17)

the number of hyperplanes that separate A and wA.

For example, `(π∨) = 0 for all π∨ ∈ Π∨, and `(s∨i ) = 1 for all simple reflections.

The Bruhat order on W∨ is defined as follows: v ≤ w in the Bruhat order if a reduced

expression for v is a substring of a reduced expression for w.

For v < w in W∨ such that w = vs∨i1 · · · s∨ir and `(w) = `(v) + r, the set of positive

coroots

L(v, w) = {vα∨i1 , . . . , vs∨i1· · · s∨ir−1

α∨ir} (2.18)

index the hyperplanes that separate the alcoves vA and wA. Note L(1, w) = L(w).

Figure 2.3 illustrates the alcoves of the sl3C root system, and the sequence of alcoves

associated to the element w = π∨s2s1s2s0s2s1s0 ∈ W∨. The coroots w∨i index the

hyperplanes that separate the alcove π∨A and the alcove wA.

15

Figure 2.3.

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.................................................................................................................................................................................................................................................................................................................................................................................................•

...............

...............

...............

...............

..............................................Hw∨1

............................................................

...............................................

Hw∨2

........................................................................................... ................

Hw∨3

............................................................

...............................................

Hw∨4

...............

...............

...............

...............

.............................................Hw∨5

............................................................

...............................................

Hw∨6

...............

...............

...............

...............

..............................................Hw∨7

π∨

ww∨1 = −ϕ∨ + d

w∨2 = −α∨1 + d

w∨3 = α∨2

w∨4 = −α∨1 + 2d

w∨5 = −ϕ∨ + 2d

w∨6 = −α∨1 + 3d

w∨7 = −ϕ∨ + 3d

Coset representatives

Let w0 ∈ W0 be the unique longest element in the finite Weyl group W0. For µ ∈ h∗Z, let

µ+ denote the unique dominant weight in the orbit, so that µ− = w0µ+ is the unique

antidominant weight. Let vµ be the shortest element of W0 such that

vµµ = µ−, (2.19)

and define

mµ = xµv−1µ ∈ W∨. (2.20)

By [30, (2.4.5)], mµ is the unique shortest element in the left coset xµW0.

Since the unique shortest element in the right coset W0xµ corresponds to an alcove

16

in the dominant chamber, then there is a bijection

{mµA | µ ∈ h∗Z} ←→ {alcoves in the dominant chamber}

mµ ↔ m−1µ .

(2.21)

Lemma 2.4. If µ is a dominant weight, then m−1µ = m−w0µ, where w0 ∈ W0 is the

longest element.

Proof. By definition, m−1µ = vµx

−µ = x−vµµ = x−w0µvµ. The element v−1µ satisfies

v−1µ (−w0µ) = v−1

µ (−vµµ) = −µ, (2.22)

so it remains to show that v−1µ is the shortest element in W0 satisfying (2.22). Suppose

there exists z−1 ∈ W0 with `(z) < `(vµ) that satisfies (2.22). Then z−1(−w0µ) = −µ

if and only if zµ = w0µ = µ−, contradicting the definition of vµ. Therefore, m−1µ =

x−w0µ(v−1µ )−1 = m−w0µ.

Figure 2.5 illustrates the bijection (2.21) for the sl2C root system.

Figure 2.5.

•

left cosetrepresentatives:

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

m0 m2ω m4ωm−2ωm−4ω

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

mω m3ωm−ωm−3ω m5ω

•

right cosetrepresentatives:

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

m−10 m−12ω m−14ωm−1−2ω m−1−4ω

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

..........................

m−1ω m−13ωm−1−ω m−1−3ω m−15ω

The group Π∨

The minuscule weights are the fundamental weights ωj that satisfy 〈ωj, α∨〉 ≤ 1 for

α∨ ∈ R∨+. In other words, these are the fundamental weights which are contained in the

17

closure of the fundamental alcove. Let

J = {j | ωj ∈ h∗Z is a minuscule weight} ∪ {0}.

For example, every fundamental weight is a minuscule weight in the Type slnC root

systems. In the Type sp2nC root systems, only the first fundamental weight ω1 = ε1 is

a minuscule weight.

Let π∨0 = 1, and for j ∈ J\{0}, let

π∨j = mωj = xωjv−1ωj. (2.23)

By [30, (2.5.4)], the subgroup of length zero elements in W∨ is

Π∨ ={π∨j ∈ W∨ | j ∈ J

}.

For example, Π∨ ∼= Zn+1 in the Type slnC root systems, and Π∨ ∼= Z2 in the Type sp2nC

root systems.

The following facts about the elements π∨j and vωj can be found in [30] and [8].

1. vωj = w0wωj , where wωj is the longest element in the stabilizer Wωj of ωj.

2. π∨j (0) = ωj.

3. π∨j (α∨0 ) = α∨j .

4. π∨j s∨i = s∨kπ

∨j , if π∨j (α∨i ) = α∨k .

2.4 Double affine braid groups

The relevant facts about braid groups from [30] are stated here in our notation.

18

Let{qkXµ | k ∈ 1

eZ, µ ∈ h∗Z

}be the multiplicative group isomorphic to h∗Z⊕Zδ, and

write

Xµ+kδ = qkXµ. (2.24)

The extended affine Weyl group W = 〈s0, . . . , sn〉o Π acts on{qkXµ | k ∈ 1

eZ, µ ∈ h∗Z

}by conjugation (see Equation (2.9)), so that

w(µ+ kδ) = wµ+ kδ, for w ∈ W0,

yλ∨(µ+ kδ) = µ− 〈µ, λ∨〉δ + kδ, for λ∨ ∈ hZ.

The double affine braid group B is generated by the groups{qkXµ | k ∈ 1

eZ, µ ∈ h∗Z

}and Π, and T0, T1, . . . , Tn, subject to the relations

TiTjTi · · · = TjTiTj · · · (mij factors each side), (2.25)

TiXµ = XµTi if 〈µ, α∨i 〉 = 0 for i = 0, . . . , n, (2.26)

TiXµTi = Xsiµ if 〈µ, α∨i 〉 = 1 for i = 0, . . . , n, (2.27)

πTiπ−1 = Tj if παi = αj for π ∈ Π, (2.28)

πXµπ−1 = Xπµ for π ∈ Π, (2.29)

where we use the notation 〈µ, α∨0 〉 = 〈µ,−ϕ∨〉. Since the conjugation action (2.9) of

W = W0 nY on the lattice h∗Z⊕ 1eZδ fixes δ, then q1/e is a central element of B. See [30,

Sec 3.4].

For w ∈ W with a reduced expression w = πjsi1 · · · sir where π ∈ Π and sij ∈

〈s0, . . . , sn〉, define

Tw = πjTi1 · · ·Tir . (2.30)

By relations (2.25) and (2.28) in the double affine braid group, the element Tw is inde-

pendent of the choice of a reduced word for w (also see [30, (3.1.1)]).

19

Identify the reduced expression w = πjsi1 · · · sir ∈ W with the minimal path from A

to wA (in the alcove picture dual to the one described in Section 2.3) via the sequence

of alcoves πjA, πjsi1A, . . . , πjsi1 · · · sirA in hZ, and define

Y w = πjTε1i1· · ·T εrir , where εk =

+1, if the kth step of p is

− +

................

................

................

................

.

....................................................................... .............. ,

−1, if the kth step of p is− +

................

................

................

................

.

..................................................................................... ,

(2.31)

with respect to the periodic orientation of the hyperplanes (see Section 2.3).

Also identify the reduced expression w∨ = π∨j s∨i1· · · s∨ir ∈ W

∨ with the minimal path

from A to w∨A via the sequence of alcoves π∨j A, π∨j s∨i1A, . . . , π∨j s

∨i1· · · s∨irA in h∗R (see

Figure 2.3 for example), and define

Xw = π∨j (T∨i1)ε1 · · · (T∨ir )εr , where εk =

+1, if the kth step of p is

− +

................

................

................

................

.

..................................................................................... ,

−1, if the kth step of p is− +

................

................

................

................

.

....................................................................... .............. ,

(2.32)

with respect to the periodic orientation of the hyperplanes (see Section 2.3).

For example, given the maximal root θ, maximal short root ϕ, πj ∈ Π, and π∨j ∈ Π∨,

Y θ∨ = T0Tsθ , Y ω∨j = πjTvω∨j

, Xϕ = (T∨0 )−1T−1sϕ , Xωj = π∨j T

−1

v−1ωj

. (2.33)

Let T∨i = Ti for i = 1, . . . , n. The following Theorem was discovered by Cherednik [5,

Theorem 2.2], and proved in [17, Theorem 4.10], [18, Theorem 2.2], and [30, (3.5.1)].

Theorem 2.6. (Duality) The double affine braid group B is generated by the groups

20

{qkY λ∨ | k ∈ Z, λ∨ ∈ hZ

}and Π∨, and T∨0 , T

∨1 , . . . , T

∨n , subject to the relations

T∨i T∨j T∨i · · · = T∨j T

∨i T∨j · · · (m∨ij factors each side), (2.34)

(T∨i )−1Y λ∨ = Y λ∨(T∨i )−1 if 〈αi, λ∨〉 = 0 for i = 0, . . . , n, (2.35)

(T∨i )−1Y λ∨(T∨i )−1 = Y s∨i λ∨

if 〈αi, λ∨〉 = 1 for i = 0, . . . , n, (2.36)

π∨T∨i (π∨)−1 = T∨j if π∨α∨i = α∨j for π∨ ∈ Π∨, (2.37)

π∨Y λ∨(π∨)−1 = Y π∨λ∨ for π∨ ∈ Π∨, (2.38)

where we use the notation 〈α0, λ∨〉 = 〈−ϕ, λ〉.

2.5 Double affine Hecke algebras

Let K be a field. Fix t0, t1, . . . , tn ∈ K such that ti = tj if si and sj are conjugate in W .

Further, for α ∈ R and k ∈ 1eZ, define tα+kδ = ti if α = wαi for some w ∈ W .

The double affine Hecke algebra H is the quotient of the group algebra KB of the

double affine braid group by the relations

T 2i = (t

1/2i − t

−1/2i )Ti + 1, for 0 ≤ i ≤ n. (2.39)

By [30, (4.7.5)], a K(q1/e)-basis for H is

{XµTwY

λ∨ | µ ∈ h∗Z, w ∈ W0, λ∨ ∈ hZ

},

Proposition 2.7. [25, Proposition 3.6], [30, (4.2.4)] Assume 〈αi, hZ〉 = Z. Let λ∨ ∈ hZ

and i = 0, . . . , n. Then

T∨i Yλ∨ − Y s∨i λ

∨T∨i = (t

1/2i − t

−1/2i )

Y λ∨ − Y s∨i λ∨

1− Y −α∨i, (2.40)

21

Proof. If Equation (2.40) holds for coweights λ∨ and µ∨, then

TiYλ∨±µ∨−Y s∨i (λ∨±µ∨)Ti =

(TiY

λ∨ − Y s∨i λ∨Ti

)Y ±µ

∨+ Y s∨i λ

∨(TiY

±µ∨ − Y ±s∨i µ∨Ti)

=(t1/2i − t

−1/2i

) (Y λ∨ − Y s∨i λ∨)Y ±µ

∨+ Y s∨i λ

∨ (Y ±µ

∨ − Y ±s∨i µ∨)

1− Y −α∨i

=(t1/2i − t

−1/2i

) Y λ∨±µ∨ − Y s∨i (λ∨±µ∨)

1− Y −α∨i,

and the equation holds for λ∨ ± µ∨ as well, so it suffices to prove that (2.40) holds for

the generators λ∨ of hZ.

The assumption that 〈αi, hZ〉 = Z implies that the generators λ∨ of hZ satisfy

〈αi, λ∨〉 = 0 or 1.

If 〈αi, λ∨〉 = 0, then (2.40) reduces to TiYλ∨ = Y λ∨Ti, which is the relation (2.35).

If 〈αi, λ∨〉 = 1, then siλ∨ = λ∨ − α∨i , and the relation (2.36) gives

0 = TiYλ∨ − Y s∨i λ

∨Ti = TiY

λ∨ − Y s∨i λ∨(Ti − (t

1/2i − t

−1/2i )

)= TiY

λ∨ − Y s∨i λ∨Ti − (t

1/2i − t

−1/2i )(−Y λ∨−α∨i )

1− Y −α∨i1− Y −α∨i

= TiYλ∨ − Y s∨i λ

∨Ti − (t

1/2i − t

−1/2i )

Y λ∨ − Y s∨i λ∨

1− Y −α∨i.

Remark 2.8. When working with the nonreduced root systems of Type C∨nCn, the case

〈αi, hZ〉 = 2Z arises, and the necessary modification to Proposition 2.7 is

TiYλ∨ − Y s∨i λ

∨Ti = ((t

1/2i − t

−1/2i ) + (t

1/20 − t−1/2

0 )Y −α∨i )Y λ∨ − Y s∨i λ

∨

1− Y −2α∨i. (2.41)

This reduces to (2.40) if t0 = ti. To prove (2.41), it suffices to consider the case λ∨ = α∨i .

See [30, (4.7.3)].

22

Proposition 2.9. [30, (4.7.2)] Let µ ∈ h∗Z ⊕ 1eZδ and i = 0, . . . , n. Then

TiXµ −XsiµTi =

(t

1/2i − t

−1/2i )

Xµ −Xsiµ

1−Xαiif 〈h∗Z, α∨i 〉 = Z,

((t1/2i − t

−1/2i ) + (t

1/20 − t−1/2

0 )Xαi)Xµ −Xsiµ

1−X2αi, if 〈h∗Z, α∨i 〉 = 2Z.

(2.42)

Proof. Similar to the proof of Proposition 2.7, if (2.42) holds for the weights λ and µ,

then it holds for λ±µ as well, so it suffices to prove that (2.42) holds for the generators

µ of h∗Z. Use the relations (2.26), (2.27), and (2.39) of the double affine Hecke algebra

to complete the proof.

Intertwining operators

The elements

T∨0 = (XϕTsϕ)−1 and Y −α∨0 = qY ϕ∨ ,

were previously defined in (2.33). Let T∨i = Ti for i = 1, . . . , n. The intertwining

operators are

τ∨i = T∨i +t−1/2i − t1/2i

1− Y −α∨i= (T∨i )−1 +

(t−1/2i − t1/2i )Y −α

∨i

1− Y −α∨i, for 0 ≤ i ≤ n,

π∨j = XωjTv−1ωj, for j ∈ J.

For w ∈ W∨ with a reduced expression w = π∨j s∨i1· · · s∨ir , define

τ∨w = π∨j τ∨i1· · · τ∨ir . (2.43)

By [30, (5.6.4), (5.10.13)], τ∨w is independent of the choice of a reduced word for w.

Moreover, for i = 0, . . . , n, and w ∈ W∨,

τ∨i τ∨w =

τ∨s∨i w

, if `(s∨i w) > `(w),

(τ∨i )2τ∨s∨i w, if `(s∨i w) < `(w),

(2.44)

23

where

(τ∨i )2 =

(1− t−1

i Y −α∨i

1− Y −α∨i

)(1− tiY −α

∨i

1− Y −α∨i

), for i = 0, 1 . . . , n, (2.45)

since

(τ∨i )2 = τ∨i

(T∨i +

t−1/2i − t1/2i

1− Y −α∨i

)= τ∨i T

∨i −

(t−1/2i − t1/2i )Y −α

∨i

1− Y −α∨iτ∨i

=

((T∨i )−1 +

(t−1/2i − t1/2i )Y −α

∨i

1− Y −α∨i

)T∨i −

(t−1/2i − t1/2i )Y −α

∨i

1− Y −α∨i

(T∨i +

t−1/2i − t1/2i

1− Y −α∨i

)

= 1− (t−1/2i − t1/2i )Y −α

∨i

1− Y −α∨it−1/2i − t1/2i

1− Y −α∨i=

(1− t−1

i Y −α∨i

1− Y −α∨i

)(1− tiY −α

∨i

1− Y −α∨i

).

Proposition 2.10. [30, (5.10.5), (5.10.6), (5.10.11)] For w ∈ W∨, and λ∨ ∈ hZ, the

intertwiners satisfy

τ∨wYλ∨ = Y wλ∨τ∨w . (2.46)

Proof. This follows from Equations (2.40) and (2.38).

Symmetrizers and antisymmetrizers

Let H0 be the subalgebra of H generated by T1, . . . , Tn. Let w0 ∈ W0 the longest element.

The symmetrizer and antisymmetrizer are, respectively,

10 =∑w∈W0

t−1/2w0w

Tw, (2.47)

ε0 =∑w∈W0

(−1)`(w)t1/2w0wTw. (2.48)

For i = 1, . . . , n, the elements

1i = t−1/2i + Ti = t

1/2i + T−1

i , εi = t1/2i − Ti = t

−1/2i − T−1

i ,

24

are the rank one analogues of 10 and ε0, and they satisfy

Ti1i = 1iTi = t1/2i 1i, 12

i = (t1/2i + t

−1/2i )1i, 1iεi = 0,

Tiεi = εiTi = −t−1/2i εi, ε2

i = (t1/2i + t

−1/2i )εi, εi1i = 0.

(2.49)

The Poincare polynomial of W0 is

W0(t) =∑w∈W0

tw. (2.50)

Note that if ti = t for i = 0, . . . , n, then W0(t) =∑

w∈W0t`(w).

Proposition 2.11. [30, (5.5.17)] For i = 1, . . . , n,

1. Ti10 = t1/2i 10 = 10Ti, and Tiε0 = −t−1/2

i ε0 = ε0Ti,

2. 120 = t

−1/2w0 W0(t)10, and ε2

0 = t−1/2w0 W0(t)ε0,

3. 10ε0 = 0, and ε010 = 0.

Proof. 1. By the quadratic relation (2.39),

10Ti =∑w∈W0

t−1/2w0w

TwTi =∑

w:wsi<w

t−1/2w0wsi

Tw +∑

w:wsi>w

t−1/2w0wsi

TwT2i

= t−1/2i

∑w:wsi<w

t−1/2w0w

Tw + (t1/2i − t

−1/2i )

∑w:wsi>w

t−1/2w0wsi

TwTi +∑

w:wsi>w

t−1/2w0wsi

Tw

= t−1/2i

∑w:wsi<w

t−1/2w0w

Tw + (t1/2i − t

−1/2i )

∑w:w>wsi

t−1/2w0w

Tw + t1/2i

∑w:wsi>w

t−1/2w0w

Tw

= t1/2i

∑w:w>wsi

t−1/2w0w

Tw + t1/2i

∑w:wsi>w

t−1/2w0w

Tw

= t1/2i 10.

A similar calculation with multiplication by Ti on the left and using (2.39) to expand

T 2i yields Ti10 = t

1/2i 10. The proof for Tiε0 = −t−1/2

i ε0 = ε0Ti is completely analogous.

25

2. It follows from 1. that

120 =

∑w∈W0

t−1/2w0w

Tw10 =∑w∈W0

t−1/2w0w

t1/2w 10 = t−1/2w0

∑w∈W0

tw10,

ε20 =

∑w∈W0

(−1)`(w)t1/2w0wTwε0 =

∑w∈W0

t1/2w0wt−1/2w ε0 = t1/2w0

∑w∈W0

t−1w ε0.

Since tw0wtw = tw0 , then t1/2w0 W0(t−1) = t

−1/2w0 W0(t).

3. It follows from 1. and (2.49) that

(t−1/2i + t

1/2i )210ε0 = 10(t

−1/2i + Ti) · (t1/2i − Ti)ε0 = 101iεiε0 = 0.

The proof for ε010 = 0 is completely analogous.

Remark 2.12. More generally (see [30, Section 5.5]), let ε be a linear character for W0,

so that ε(si) = ±1 for i = 1, . . . , n. For the simply-laced root systems of Types ADE,

ε is either the trivial character or the sign character. For the remaining root systems,

there are four distinct linear characters. Define

t(ε)i =

ti, if ε(si) = +1,

−t−1i , if ε(si) = −1,

and for w ∈ W0, let t(ε)w = t

(ε)i1· · · t(ε)ir if w = si1 · · · sir is a reduced expression. One can

define

Uε =(t(ε)w0

)−1/2 ∑w∈W0

(t(ε)w)1/2

Tw, (2.51)

so that Utriv = 10 and Usgn = ε0.

2.6 The polynomial representation

The affine Hecke algebra H is the subalgebra of the double affine Hecke algebra H

generated by T0, . . . , Tn and Π. It is the quotient of the group algebra of the braid group

26

of W by the relations (2.39). A basis for H is {TwY λ∨ | w ∈ W0, λ∨ ∈ hZ} (see [30,

(4.2.7)]).

Let K1 be the H-module given by

π1 = 1, Ti1 = t1/2i 1, for π ∈ Π and i = 0, . . . , n.

The polynomial representation of H is

K[X]1 = IndHH1, (2.52)

and has basis {Xµ1 | µ ∈ h∗Z}.

For an affine coroot β∨ + jd, define shift and height

qsh(β∨+jd) = q−j, and tht(β∨+jd) =∏α∈R+

t12〈α,β∨〉

α , (2.53)

so that

Y β∨+jd1 = q−jY β∨1 = q−j∏α∈R+

t12〈α,β∨〉

α 1 = qsh(β∨+jd)tht(β∨+jd)1. (2.54)

If tα = t for all α ∈ R+, then tht(β∨+jd) = t〈ρ,β∨〉, where

ρ =1

2

∑α∈R+

α∨. (2.55)

Polynomial rings

The action of the Weyl group W0 on K[X]1 is given by

siXµ = Xsiµ, for i = 1, . . . , n,

and the subspace of W0-invariant polynomials is denoted by

K[X]W01 = {f1 ∈ K[X]1 | wf1 = f1 for all w ∈ W0}.

27

Assume 〈h∗Z, α∨i 〉 = Z, where i = 0, . . . , n. By Proposition 2.9, the operators Ti act

on K[X]1 by

Tif1 = t1/2i (sif)1 +

(t1/2i − t

−1/2i

) f − sif1−Xαi

1, (2.56)

where Xα0 = qX−ϕ.

Let

∆ =∏α∈R+

(t1/2α Xα/2 − t−1/2

α X−α/2)

= t1/2w0Xρ

∏α∈R+

(1− t−1α X−α). (2.57)

Proposition 2.13.

1. 10H1 = {f1 ∈ K[X]1 | Tif1 = t1/2i f1 for i = 1, . . . , n} = K[X]W01,

2. ε0H1 = {f1 ∈ K[X]1 | Tif1 = −t−1/2i f1 for i = 1, . . . , n} = ∆K[X]W01.

Proof. 1. Let f1 ∈ K[X]1. By Proposition 2.11, Ti10f1 = t1/2i 10f1 for i = 1, . . . , n. So

10H1 ⊆ {f | Tif = t1/2i f}.

Suppose Tif1 = t1/2i f1 for i = 1, . . . , n. Then

0 =(t1/2i − Ti

)f1 = t

1/2i f1− t1/2i (sif)1− t

1/2i − t

−1/2i

1−Xαi(1− si)f1

=

(t1/2i −

t1/2i − t

−1/2i

1−Xαi

)(1− si)f1,

so sif1 = f1 for every i = 1, . . . , n. So {f | Tif = t1/2i f} ⊆ K[X]W01.

Lastly, let f1 ∈ K[X]W01. Since Ti commutes with any f1 ∈ K[X]W01, then

10f1 = f101 = t−1/2w0

W0(t)f1,

and f1 ∈ 10H1. Therefore, K[X]W01 ⊆ 10H1.

2. Let f1 ∈ K[X]1. By Proposition 2.11, Tiε0f1 = −t−1/2i ε0f1 for i = 1, . . . , n. So

ε0H1 ⊆ {f | Tif = −t−1/2i f}.

28

Conversely, suppose f1 satisfies Tif1 = −t−1/2i f1 for all i = 1, . . . , n. Then

ε0f1 =∑w∈W0

(−1)`(w)t1/2w0wTwf1 =

∑w∈W0

(−1)`(w)t1/2w0w(−t−1/2

w )f1 = t−1/2w0

W0(t)f1,

so {f | Tif = −t−1/2i f} ⊆ ε0H1.

Next, suppose Tif1 = −t−1/2i f1 for i = 1, . . . , n. Let ∆i = t

1/2i Xαi/2 − t−1/2

i X−αi/2,

and g1 = f/∆1. Then

0 = (Ti + t−1/2i )f1 = (Ti + t

−1/2i )∆g1

=

((si∆)Ti + (t

1/2i − t

−1/2i )

∆− si∆1−Xαi

+ t−1/2i ∆

)g1

=

((si∆)Ti − (t

1/2i − t

−1/2i )

(si∆

si∆i

)∆i − si∆i

1−Xαi+ t−1/2i ∆

)g1

=

((si∆)Ti − (ti − t−1

i )

(si∆

si∆i

)X−αi/2 + t

−1/2i ∆

)g1

=

((si∆)Ti − t1/2i si∆i

(si∆

si∆i

)− t−1/2

i ∆i

(si∆

si∆i

)+ t−1/2i ∆

)g1

=(

(si∆)Ti − t1/2i si∆)g1

= (si∆)(Ti − t1/2i )g1,

so that Tig1 = t1/2i g1 for i = 1, . . . , n. Thus g1 ∈ K[X]W01. Furthermore,

1

∆f1 = g1 = (w0g)1 =

1

w0∆(w0f) 1.

Since ∆ and w0∆ have no common factors, then ∆ divides f1, and g1 ∈ K[X]1. Hence

{f | Tif = −t−1/2i f} ⊆ ∆K[X]W01.

Conversely, suppose f1 = ∆g1 where g1 ∈ K[X]W01. Then

Tif1 = Ti∆g1 = (si∆)(Ti − t1/2i )g1− t−1/2i f1 = −t−1/2

i f1,

for i = 1, . . . , n, so ∆K[X]W01 ⊆ {f | Tif = −t−1/2i f}.

29

2.7 Alcove walks

Fix a reduced factorization of w = π∨j s∨i1· · · s∨ir ∈ W

∨. An alcove walk of type ~w begin-

ning at z is a sequence of steps in the alcove picture, where for k = 0, . . . , n, a step of

type s∨k is one of the following:

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

......................................................................................... ..............

v vsk

s∨k -crossing,

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.........................................................................................................................

v vsk

s∨k -folding.

(2.58)

In addition, a step of type ~π∨ for π∨ ∈ Π∨ is a “change of sheets” from the alcove v to

the alcove vπ∨j . See Figure 2.1.

Let

Γ(~w, z) be the set of alcove walks of type ~w beginning in z. (2.59)

There are 2r walks in Γ(~w, z), since each step can be either a crossing or a folding. For

a walk p ∈ Γ(~w, z), let

p∨k be the positive coroot such that Hp∨k separates the alcoves v and vs∨ik , (2.60)

where v is the alcove where k − 1th step of p ended. Also let

b∨k = s∨irs∨ir−1· · · s∨ik+1

α∨ik , for k = 1, . . . , r, (2.61)

so that {b∨r , . . . , b∨1 } = L(w−1) as defined in (2.15). See Example 2.14.

Example 2.14. Let w = (s1s∨0 )4 = x−8ω. The following is an alcove walk p ∈ Γ(~w, 1)

(on sheet 1) in the Type sl2C alcove picture. See Section 5.1 for more details.

•

H−α∨+2dHα∨Hα∨+2d H−α∨+d H−α∨+3dHα∨+d H−α∨+4d H−α∨+5d

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

+− +− +− +− +− +− +− +−

1

x4ωs1...................................................................................................................................................................................................................................................................................... ................ ........................................................................................................................... ................ ........................................................................................................................... ................ ........................................................................................................................... ................ ...................................................................................................................................................................................................................................................................................................... ................ ........................................................................................................................... ................

30

This walk of length 8 has type (s1, s∨0 , s1, s

∨0 , s1, s

∨0 , s1, s

∨0 ). The coroots p∨k are

p∨1 , . . . , p∨8 = α∨, α∨ + d, α∨, −α∨ + d, −α∨ + 2d, −α∨ + 3d, −α∨ + 2d, −α∨ + 3d.

The coroots b∨k are associated to the following walk:

•.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................

.................................................................

.................................................................

.................................................................

.................................................................

.................................................................

.................................................................

.................................................................

.................................................................

.................................................................

.................................................................

.................................................................

+−

w−1

1Hb∨8 Hb∨7 Hb∨6 Hb∨5 Hb∨4 Hb∨3 Hb∨2 Hb∨1

...................................................................................................................... ...................................................................................................................... ...................................................................................................................... ...................................................................................................................... ...................................................................................................................... ...................................................................................................................... ...................................................................................................................... ......................................................................................................................

where b∨k = −α∨ + (9− k)d for k = 1, . . . , 8. �

Positive and negative steps are defined as follows:

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

......................................................................................... ..............+−

v vsj

positive crossing,

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......................................................................................................+−

vsj v

negative crossing,

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

........................................................................................................... ..............+−

vsj v

positive folding,

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.........................................................................................................................+−

v vsj

negative folding.

(2.62)

Moreover,

a step (of type s∨j ) is long if `(vs∨j ) > `(v), and it is short if `(vs∨j ) < `(v). (2.63)

That is, a crossing is long if it points away from the fundamental alcove and is short

if it points towards the fundamental alcove, while a folding is long if it folds towards

the fundamental alcove and is short if folds away from the fundamental alcove. In

Example 2.14, the first step of the walk p is a negative crossing, the second and seventh

steps are positive folds, the third, fourth, fifth and eighth step are positive crossings,

and the sixth step is a negative fold. Also, the second and sixth steps are long folds,

while the seventh step is a short fold.

31

Chapter 3

Macdonald polynomials

3.1 Nonsymmetric Macdonald polynomials

Given a weight µ ∈ h∗Z, let mµ ∈ W∨ be the shortest element in the coset xµW0,

see (2.20).

Definition 3.1. Let mµ = π∨j s∨i1· · · s∨ir be a reduced expression. The nonsymmetric

Macdonald polynomial indexed by µ ∈ h∗Z is

Eµ1 = τ∨mµ1 = π∨j τ∨i1· · · τ∨ir1. (3.1)

In the literature, Eµ is often normalized so that the coefficient of the leading mono-

mial Xµ in Eµ is 1. However, from the point of view of defining Eµ via products of

intertwiners, it is more natural to choose a different normalization. By Definition 3.1,

Eµ1 = π∨j (T∨i1)ε1 · · · (T∨ir )εr1 + lower terms = XµTv−1

µ1 + lower terms,

so the coefficient of the leading monomial Xµ in Eµ is t1/2

v−1µ

, where vµ ∈ W0 is as in

Equation (2.19).

Remark 3.2. Because nonsymmetric Macdonald polynomials can be obtained by suc-

cessively applying intertwiners to the polynomial 1, the intertwiners are also called

creation operators, see [30, Section 5.10].

32

By Proposition 2.10, the nonsymmetric Macdonald polynomials are the eigenfunc-

tions of the operators {Y λ∨ | λ∨ ∈ hZ}, and by Equations (2.46) and (2.54), the eigen-

values are

Y λ∨Eµ1 = τ∨mµYm−1µ λ∨1 = τ∨mµY

vµλ∨+〈µ,λ∨〉d = q−〈µ,λ∨〉t〈ρ,vµλ

∨〉Eµ1. (3.2)

Proposition 3.3. [30, (2.8.5), (5.2.2)] The set {Eµ1 | µ ∈ h∗Z} is a basis for K[X]1.

Proof. The leading monomial of Eµ1 is Xµ1, so {Eµ1 | µ ∈ h∗Z} spans K[X]1. To see

that the nonsymmetric Macdonald polynomials are linearly independent, it suffices to

show that the eigenvalues Y λ∨Eµ1 = q−〈µ,λ∨〉t〈v

−1µ ρ,λ∨〉Eµ1 are distinct.

Since q−〈µ,λ∨〉t〈v

−1µ ρ,λ∨〉1 = Y 〈m

−1µ λ∨,ρ〉1 = Y 〈−λ

∨,mµ(−ρ)〉1, it remains to show that

mµ(−ρ) = mν(−ρ) implies µ = ν. First,

vµmµ(−ρ) = vµxµv−1

µ (−ρ) = xvµµ(−ρ) = −ρ+ µ−,

where ρ = 12

∑α∈R+

α =∑n

i=1 ωi, and µ− are antidominant weights. Hence −ρ + µ− is

the antidominant weight in the orbit of W0mµ(−ρ). Assuming that mµ(−ρ) = mν(−ρ),

then vµmµ(−ρ) = −ρ+µ− and vνmµ(−ρ) = vνmν(−ρ) = −ρ+ν− are both antidominant

weights in the same W0-orbit, so they are the equal. This implies µ− = ν− and vµ = vν ,

and therefore µ = ν.

Therefore, {Eµ1 | µ ∈ h∗Z} are eigenvectors of {Y λ∨ | λ∨ ∈ hZ} with distinct eigen-

values, and hence are linearly independent.

For w, z ∈ W∨, Γ(~w, z) is the set of alcove walks of type ~w beginning in z. Given a

walk p ∈ Γ(~w, z), let

φ(p) = {k | the kth step of p is a fold},

φ−(p) = {k | the kth step of p is a negative fold},(3.3)

33

e(p) ∈ W∨ be the alcove where p ends. (3.4)

The following gives an expansion of a product of monomials and intertwiners in terms

of monomials.

Theorem 3.4. [35, Theorem 2.2] Let z, w ∈ W∨, and fix a reduced expression w =

π∨j s∨i1· · · s∨ir . Then

Xzτ∨w =∑

p∈Γ(~w,z)

Xe(p)fp(Y ),

where as defined in Section 2.7, Γ(~w, z) is the set of walks of type ~w beginning in z,

fp(Y ) =∏k∈φ(p)

t−1/2

b∨k− t1/2b∨k

1− Y −b∨k∏

k∈φ−(p)

Y −b∨k ,

and b∨k = s∨ir · · · s∨ik+1

α∨ik , see (2.61).

Proof. Proceed by induction on `(w). If `(w) = 0 so that w ∈ Π∨, then Xzτ∨w = Xzw.

If `(w) = 1, then the base cases are

τ∨i = T∨i +t−1/2i − t1/2i

1− Y −α∨i= (T∨i )−1 +

(t−1/2i − t1/2i )Y −α

∨i

1− Y −α∨i, for i = 0, . . . , n.

Let v = π∨j s∨i1· · · s∨ir−1

, and let p ∈ Γ(~v, z). Let p1, p2 ∈ Γ(~vs∨ir , z) be the two extensions

of p by a crossing and a folding of type s∨ir respectively. Let w = vs∨ir . By induction, a

term in Xzτ∨w = Xzτ∨v τ∨ir is

Xe(p)fp(Y )τ∨ir = Xe(p)τ∨irsir(fp(Y ))

=

Xe(p)

(T∨ir +

t−1/2ir− t1/2ir

1− Y −α∨ir

)sir(fp(Y )), if Xzsir = XzT∨ir ,

Xe(p)

(T∨ir +

(t−1/2ir− t1/2ir

)Y −α∨i

1− Y −α∨ir

)sir(fp(Y )), if Xzsir = Xz(T∨ir )

−1,

= Xe(p1)fp1(Y ) +Xe(p2)fp2(Y ). �

34

Define the weight wt(p) ∈ h∗Z and final direction d(p) ∈ W0 of a walk p by

Xe(p) = Xwt(p)Td(p). (3.5)

Also, let tw = ti1 · · · tir if w = si1 · · · sir ∈ W0 is a reduced expression. The following

gives a change of basis formula from the nonsymmetric Macdonald polynomials to the

monomials.

Corollary 3.5. [35, Theorem 3.1] Let µ ∈ h∗Z, and fix a reduced expression for the mini-

mal length representative mµ = π∨j s∨i1· · · s∨ir ∈ W

∨ of the coset xµW0. The nonsymmetric

Macdonald polynomial indexed by µ is

Eµ1 =∑

p∈Γ(~mµ,1)

t1/2d(p)fpX

wt(p)1,

where Γ(~mµ, 1) is the set of walks of type ~mµ beginning in the fundamental alcove,

fp =∏k∈φ(p)

t−1/2

b∨k− t1/2b∨k

1− qsh(−b∨k )tht(−b∨k )

∏k∈φ−(p)

qsh(−b∨k )tht(−b∨k ), (3.6)

and b∨k = s∨ir · · · s∨ik+1

α∨ik , see (2.61).

Proof. Since Eµ1 = τ∨mµ1, then the result follows from Theorem 3.4 by setting w = mµ

and z = 1, and using

Y −b∨k 1 = qsh(−b∨k )tht(−b∨k )1, Xe(p)1 = Xwt(p)Td(p)1 = t

1/2d(p)X

wt(p)1,

where sh(−b∨k ) and ht(−b∨k ) are as defined in (2.53).

See Section 5.2 for examples of nonsymmetric Macdonald polynomials associated to

the root system of sl3C.

35

Fix a reduced expression z = s∨ir · · · s∨i1

(π∨j )−1 so that Xz = (T∨ir )εr · · · (T∨i1)ε1(π∨j )−1

where εk ∈ {+1,−1}. If b is the unfolded walk of type ~z−1 beginning in z, then

εk =

+1, if the kth step of b is positive,

−1, if the kth step of b is negative,

(3.7)

For a walk p ∈ Γ(~z−1, w−1), let

ξs(p) = {k | the kth step of p is a short crossing},

φl(p) = {k | the kth step of p is a long fold},(3.8)

ψ(p) =

k∣∣∣∣∣ the kth step of p is a long fold and εk = −1

or the kth step of p is a short fold and εk = +1

. (3.9)

The following gives an expansion of a product of monomials and intertwiners in terms

of intertwiners.

Theorem 3.6. Let z, w ∈ W∨, and fix a reduced expression z = s∨ir · · · s∨i1

(π∨j )−1. Then

Xzτ∨w =∑

p∈Γ(~z−1,w−1)

τ∨e(p)−1gp(Y )np(Y ),

where as defined in Section 2.7, Γ(~z−1, w−1) is the set of walks of type ~z−1 beginning in

w−1,

np(Y ) =∏

k∈ξs(p)

1− t−1ikY −p

∨k

1− Y −p∨k1− tikY −p

∨k

1− Y −p∨k,

gp(Y ) = (−1)|φl(p)|∏k∈φ(p)

t−1/2ik− t1/2ik

1− Y −p∨k∏

k∈ψ(p)

Y −p∨k ,

and p∨k is defined so that the kth step of p crosses or folds against the hyperplane Hp∨k ,

see (2.60).

36

Proof. We proceed by induction on `(z). If `(z) = 0 so that z = (π∨j )−1 ∈ Π∨, then the

only walk p in Γ(~z−1, w−1) is a change between sheets from the alcove w−1 to the alcove

w−1z−1. So in this case, Xzτ∨w = τ∨zw = τ∨e(p)−1 .

Let v = s∨ir−1· · · s∨i1(π∨j )−1, and let p ∈ Γ(~v−1, w−1), with

np(Y ) =∏

k∈ξs(p)

1− t−1ikY −p

∨k

1− Y −p∨k1− tikY −p

∨k

1− Y −p∨k,

gp(Y ) = (−1)|φl(p)|∏k∈φ(p)

t−1/2ik− t1/2ik

1− Y −p∨k∏

k∈ψ(p)

Y −p∨k .

Let p1, p2 ∈ Γ(~v−1s∨ir , w−1) be the two extensions of p by a crossing and a folding of type

ir, respectively.

Let z = s∨irv. By induction, a term in Xzτ∨w = (T∨ir )εrXvτ∨w is

(T∨ir)εr

τ∨e(p)−1gp(Y )np(Y ) =

τ∨ir − (t−1/2ir− t1/2ir

)(Y −α

∨ir

) 12

(1−εr)

1− Y −α∨ir

τ∨e(p)−1gp(Y )np(Y )

=

τ∨irτ∨e(p)−1 − τ∨e(p)−1

(t−1/2ir− t1/2ir

)(Y −e(p)α∨ir

) 12

(1−εr)

1− Y −e(p)α∨ir

gp(Y )np(Y ).

The first term

τ∨irτ∨e(p)−1gp(Y )np(Y )

=

τ∨e(p1)−1gp1(Y )np1(Y ), if `(s∨ire(p)−1) > `(s∨ire(p)−1),

τ∨e(p1)−1

1−t−1irY−e(p)α∨ir

1−Y −e(p)α∨ir

1−tirY−e(p)α∨ir

1−Y −e(p)α∨irgp1(Y )np(Y ), if `(s∨ire(p)−1) < `(s∨ire(p)−1),

= τ∨e(p1)−1gp1(Y )np1(Y ),

37

and the second term

−τ∨e(p)−1

(t−1/2ir− t1/2ir

)(Y −e(p)α∨ir

) 12

(1−εr)

1− Y −e(p)α∨irgp(Y )np(Y )

=

−τ∨e(p2)−1

(t−1/2ir− t1/2ir

)(Y −p

∨r) 1

2(1−εr)

1− Y −p∨rgp(Y )np2(Y ), if `(s∨ire(p)−1) > `(s∨ire(p)−1),

+τ∨e(p2)−1

(t−1/2ir− t1/2ir

)(Y −p

∨r) 1

2(1+εr)

1− Y −p∨rgp(Y )np2(Y ), if `(s∨ire(p)−1) < `(s∨ire(p)−1),

= τ∨e(p2)−1gp2(Y )np2(Y ). �

The following gives a change of basis formula from the monomial basis to the non-

symmetric Macdonald basis.

Corollary 3.7. Let µ ∈ h∗Z, and fix a reduced expression xµ = s∨ir · · · s∨i1

(π∨j )−1. The

monomial Xµ as a linear combination of nonsymmetric Macdonald polynomials is

Xµ1 =∑

p∈ΓC( ~xµ−1,1)

gpnpE$(p)1,

where ΓC( ~xµ−1, 1) is the set of walks of type (π∨j , i1, . . . , ir) beginning in the fundamental

alcove and contained in the dominant chamber C,

np =∏

k∈ξs(p)

1− qsh(−p∨k )tht(−p∨k )t−1ik

1− qsh(−p∨k )tht(−p∨k )

1− qsh(−p∨k )tht(−p∨k )tik1− qsh(−p∨k )tht(−p∨k )

,

gp = (−1)|φl(p)|∏k∈φ(p)

t−1/2ik− t1/2ik

1− qsh(−p∨k )tht(−p∨k )

∏k∈ψ(p)

qsh(−p∨k )tht(−p∨k ),

and $(p) ∈ h∗Z is defined by e(p)−1 = m$(p).

Proof. We shall show that if the walk leaves the dominant chamber, then it does not

contribute to the sum. For p ∈ Γ( ~xµ−1, 1), identify τ∨e(p)−1np(Y ) with the sequence of

crossing steps of p. That is,

τ∨e(p)−1np(Y ) = τ∨ich· · · τ∨ic1 (π∨j )−1,

38

where the cjth step of p is a crossing for j = 1, . . . , h.

The bijection between left and right cosets gives a bijection between minimal length

(left) coset representatives and alcoves in the dominant chamber (minimal length right

coset representatives) via taking inverses:

W∨/W0 ←→ W0\W∨

mµ ↔ m−1µ .

If the walk p exits the dominant chamber at the kth crossing, then(s∨ick· · · s∨ic1 (π∨j )−1

)−1

is not a minimal length coset representative, and

τ∨e(p)−1gp(Y )np(Y )1 = (e(p)−1 · gp(Y )) τ∨ich· · ·(τ∨ick· · · τ∨ic1 (π∨j )−11

)= 0.

The result follows from Theorem 3.6 by setting z = xµ, w = 1, and using

Y −p∨k 1 = qsh(−p∨k )tht(−p∨k )1, and τ∨e(p)−11 = E$(p)1,

where sh(−b∨k ) and ht(−b∨k ) are as defined in (2.53).

See Example 5.11 for an illustration of the expansion of X−α21 in the Type sl3C

nonsymmetric Macdonald polynomial basis.

Remark 3.8.

(a) When a walk is contained in the dominant chamber, a short crossing is equivalent

to a negative crossing.

(b) If µ is a dominant weight, then εk = −1 for all k = 1, . . . , r in Equation (3.7).

Thus ψ(p) in Equation (3.9) simplifies:

ψ(p) = {k | the kth step of p is a long fold}.

39

3.2 Symmetric Macdonald polynomials

Given a weight µ ∈ h∗Z, let mµ ∈ W∨ be the shortest element in the coset xµW0,

see (2.20).

Definition 3.9. Let mµ = π∨j s∨i1· · · s∨ir be a reduced expression. The symmetric Mac-

donald polynomial indexed by µ ∈ h∗Z is

Pµ1 = 10τ∨mµ1, (3.10)

where 10 =∑

w∈W0t−1/2w0w Tw, is the symmetrizer from (2.47). By Proposition 2.13, Pµ1

is W0-symmetric. Also define

Aµ1 = ε0τ∨mµ1, (3.11)

where ε0 =∑

w∈W0(−1)`(w)t

1/2w0wTw is the antisymmetrizer from (2.48).

Remark 3.10. More generally, for any linear character ε of W0, one can use the operator

Uε in (2.51) to define polynomials Uετ∨mµ1, so Pµ1 and Aµ1 correspond to the cases when ε

is the trivial character and sign character, respectively. For further properties of Uετ∨mµ1,

see [30, Section 5.7].

In the literature, Aµ is also denoted by Qµ ([30, (5.7.3)]). Often, Pµ is normalized

so that the coefficient of the orbit sum Mµ =∑

ν∈W0µXν in Pµ is 1, but as seen below,

Definition 3.9 gives a different normalization. Let

Wµ ⊆ W0 be the stabilizer of µ ∈ h∗Z. (3.12)

The parabolic subgroup Wµ has a unique longest element [2, (2.10)] which we denote by

wµ. Also, each coset vWµ has a unique shortest representative [2, Corollary 2.4.5], so let

W µ ⊆ W0 be the set of minimal length representatives of the cosets vWµ, (3.13)

40

and denote the unique longest element of W µ by vµ. Then every element w ∈ W0 has a

unique factorization w = vu such that v ∈ W µ and u ∈ Wµ. In particular, w0 = vµwµ.

Hence

10 =∑w∈W0

t−1/2w0w

Tw =∑v∈Wµ

t−1/2vµv Tv

∑u∈Wµ

t−1/2wµu Tu. (3.14)

Since Tuf1 = t1/2u f1 if uf1 = f1 by (2.56), then

Pµ1 = 10Eµ1 = t−1/2wµ Wµ(t)

∑v∈Wµ

t−1/2vµv TvEµ1,

where Wµ(t) =∑

u∈Wµtu is the Poincare polynomial of Wµ. Thus, the coefficient of the

orbit sum Mµ in Pµ is t−1/2wµ Wµ(t).

Proposition 3.11. For i = 0, . . . , n, and µ ∈ h∗Z,

τ∨i τ∨mµ =

τ∨s∨i mµ, if s∨i µ > µ,

τ∨mµτ∨k if s∨i µ = µ, where mµα

∨k = α∨i ,

(τ∨i )2τ∨s∨i mµ, if s∨i µ < µ.

(3.15)

Proof. The first and third cases follow from (2.44). So suppose s∨i µ = µ. Then µ lies

on the hyperplane Hα∨i . The hyperplane Hα∨i is a wall of the alcove mµA, and the alcove

s∨i mµA lies on the other side of Hα∨i . Thus s∨i mµ = mµs∨k , where mµα

∨k = α∨i by (2.15).

Remark that k 6= 0, since the alcoves mµA and s∨i mµA are in the same coset xµW0.

Proposition 3.12. [30, (5.7.2)] Let µ ∈ h∗Z and w ∈ W0 such that wmµ > mµ. Then

Pwµ1 =

∏a∨∈L(m−1

µ ,(wmµ)−1)

t−1/2a∨

1− qsh(−a∨)tht(−a∨)ta∨

1− qsh(−a∨)tht(−a∨)

Pµ1,

Awµ1 =

∏a∨∈L(m−1

µ ,(wmµ)−1)

t1/2a∨

1− qsh(−a∨)tht(−a∨)t−1a∨

1− qsh(−a∨)tht(−a∨)

Aµ1.

Moreover, if siµ = µ for some i = 1, . . . , n, and t1/2i + t

−1/2i 6= 0, then Aµ1 = 0.

41

Proof. Let w = si1 · · · sir ∈ W0 be a reduced expression. By [30, (5.5.9)], 10Ti = 10t1/2i

for i = 1, . . . , n, so

Pwµ1 = 10τ∨wτ∨mµ1 = 10

(t1/2i1

+t−1/2i1− t1/2i1

1− Y −α∨i1

)τ∨i2 · · · τ

∨irτ∨mµ1

= 10τ∨i2· · · τ∨irτ

∨mµ

(t1/2i1

+t−1/2i1− t1/2i1

1− Y −m−1µ s∨ir ···s

∨i2α∨i1

)1

= 10τ∨mµ

r∏j=1

(t1/2ij

+t−1/2ij− t1/2ij

1− Y −m−1µ s∨ir ···s

∨ij+1

α∨ij

)1

= 10τ∨mµ

∏a∨∈L(m−1

µ ,m−1µ w−1)

(t1/2a∨ +

t−1/2a∨ − t1/2a∨

1− Y −a∨

)1

= 10τ∨mµ

∏a∨∈L(m−1

µ ,m−1µ w−1)

(t1/2a∨ +

t−1/2a∨ − t1/2a∨

1− qsh(−a∨)tht(−a∨)

)1

=∏

a∨∈L(m−1µ ,m−1

µ w−1)

t−1/2a∨

1− qsh(−a∨)tht(−a∨)ta∨

1− qsh(−a∨)tht(−a∨)Pµ1.

Similarly, ε0Ti = −ε0t−1/2i for i = 1, . . . , n, so

Awµ1 = ε0τ∨wτ∨mµ1 = ε0τ

∨mµ

r∏j=1

(−t−1/2

ij+

t−1/2ij− t1/2ij

1− Y −m−1µ s∨ir ···s

∨ij+1

α∨ij

)1

= ε0τ∨mµ

∏a∨∈L(m−1

µ ,m−1µ w−1)

(−t−1/2

a∨ +t−1/2a∨ − t1/2a∨

1− qsh(−a∨)tht(−a∨)

)1

=∏

a∨∈L(m−1µ ,m−1

µ w−1)

t1/2a∨

1− qsh(−a∨)tht(−a∨)t−1a∨

1− qsh(−a∨)tht(−a∨)Aµ1.

If siµ = µ, then by Proposition 3.11, τ∨i τmµ1 = τ∨mµτ∨k 1 = 0, since k 6= 0. So

0 = ε0τ∨i τ∨mµ1 = ε0τ

∨mµ

(−t−1/2

i +t−1/2i − t1/2i

1− Y −m−1µ α∨i

)1 = ε0τ

∨mµ

(−t−1/2

i +t−1/2i − t1/2i

1− Y −α∨k

)1

= ε0τ∨mµ

(−t−1/2

i +t−1/2i − t1/2i

1− t−1i

)1 = −

(t1/2i + t

−1/2i

)Aµ1.

So Aµ1 = 0 if µ is not a regular weight.

42

Proposition 3.13. Let (h∗Z)+ be the set of dominant weights, and let (h∗Z)++ be the set

of regular dominant weights. Then

{Pµ1 | µ ∈ (h∗Z)+} is a basis for 10H1.

{Aµ1 | µ ∈ (h∗Z)++} is a basis for ε0H1.

Proof. Proposition 3.11 showed that Pν1 is a scalar multiple of Pµ1 for all ν ∈ W0µ,

and Proposition 3.3 showed that {Eµ1 | µ ∈ h∗Z} is a basis for K[X]1 = H1, therefore

{Pµ1 | µ ∈ (h∗Z)+} = {10Eµ1 | µ ∈ (h∗Z)+} spans 10H1. For µ ∈ (h∗Z)+ dominant, since

Pµ1 = t−1/2wµ Wµ(t)Mµ + lower terms,

then the {Pµ1 | µ ∈ (h∗Z)+} are linearly independent.

The proof for ε0H1 is similar to the above, the only variation being that Aµ1 = 0 if

µ is not a regular weight, by Proposition 3.11.

Let

i(p) ∈ W∨ be the alcove where p begins. (3.16)

The following gives an expansion of Pµ1 and Aµ1 in terms of monomials.

Theorem 3.14. Let µ ∈ h∗Z, and fix a reduced expression mµ = π∨j s∨i1· · · s∨ir ∈ W

∨ for

the minimal length representative of the coset xµW0.

1. [35, Theorem 3.4] The symmetric Macdonald polynomial indexed by µ is

Pµ1 =∑w∈W0

∑p∈Γ(~mµ,w)

t−1/2w0i(p)t

1/2d(p)fpX

wt(p)1,

2. and

Aµ1 =∑w∈W0

∑p∈Γ(~mµ,w)

(−1)`(i(p))t1/2w0i(p)t

1/2d(p)fpX

wt(p)1,

43

where each sum is over the walks of type ~mµ beginning in w ∈ W0, and

fp(Y ) =∏k∈φ(p)

t−1/2bk− t1/2bk

1− qsh(−b∨k )tht(−b∨k )

∏k∈φ−(p)

qsh(−b∨k )tht(−b∨k ),

for b∨k = s∨ir · · · s∨ik+1

α∨ik , see (2.61).

Proof. Since Pµ1 = 10Eµ1 =∑

w∈W0t−1/2w0w X

wτ∨mµ1, the result follows from computing

by the same method as in Corollary 3.5, where the additional scalar factors come from

10. The computation for Aµ1 = ε0Eµ1 =∑

w∈W0(−1)`(w)t

1/2w0wX

wτ∨mµ is similar.

The following proposition, implicit in [28, p.203], provides an expression for 10 in

terms of intertwiners. Its corollary is an expansion of Pµ1 and Aµ1 in terms of nonsym-

metric Macdonald polynomials.

Proposition 3.15.

10 =∑w∈W0

τ∨w∏

a∨∈L(w−1,w0)

t1/2a∨

(1− t−1

a∨Y−a∨

1− Y −a∨),

ε0 =∑w∈W0

(−1)`(w)τ∨w∏

a∨∈L(w−1,w0)

t−1/2a∨

(1− ta∨Y −a

∨

1− Y −a∨).

where L(w−1, w0) is the set of coroots indexing the hyperplanes which separate w−1A and

w0A, as defined in (2.18).

Proof. Since 10 =∑

w∈W0t−1/2w0w Tw, then 10 can be expressed in the form

∑w∈W0

τ∨wbw(Y )

where bw(Y ) is a rational function in Y . The coefficient of Tw0 in 10 is 1, hence

bw0(Y ) = 1. The other bw(Y ) can be computed by induction on the length of w. By

44

Proposition 2.11, 10Ti = 10t1/2i for i = 1, . . . , n, so∑

w∈W0

τ∨wbw(Y )t1/2i = 10t

1/2i = 10Ti =

∑w∈W0

τ∨wbw(Y )

(τ∨i −

t−1/2i − t1/2i

1− Y −α∨i

)

=∑w∈W0

τ∨wτ∨i (sibw(Y ))−

∑w∈W0

τ∨wt−1/2i − t1/2i

1− Y −α∨ibw(Y )

=∑

w:w<wsi

τ∨wsi (sibw(Y )) +∑

w:w>wsi

τ∨wsi(τ∨i )2 (sibw(Y ))−

∑w∈W0

τ∨wt−1/2i − t1/2i

1− Y −α∨ibw(Y )

=∑

w:w>wsi

τ∨w (sibwsi(Y )) + τ∨wsi(τ∨i )2 (sibw(Y ))−

∑w∈W0

τ∨wt−1/2i − t1/2i

1− Y −α∨ibw(Y ).

Let z = wsi such that z = wsi < w. Equating the coefficients of τ∨z on both sides of the

equation yields

bz(Y )t1/2i = (τ∨i )2 (sibw(Y ))− t

−1/2i − t1/2i

1− Y −α∨ibz(Y ),

and since (τ∨i )2 =

(t1/2i +

t−1/2i −t1/2i

1−Y −α∨i

)(t−1/2i − t

−1/2i −t1/2i

1−Y −α∨i

), then by induction,

bz(Y ) =

(t−1/2i − t

−1/2i − t1/2i

1− Y −α∨i

)(sibw(Y ))

=

(t−1/2i − t

−1/2i − t1/2i

1− Y −α∨i

) ∏a∨∈siL(w−1,w0)

(t−1/2a∨ − t

−1/2a∨ − t1/2a∨

1− Y −a∨

)

=∏

a∨∈L(z−1,w0)

t1/2a∨

(1− t−1

a∨Y−a∨

1− Y −a∨),

since siL (w−1, w0) ∪ {α∨i } = L (siw−1, siw0) ∪ L(siw0, w0) = L (z−1, w0) .

The proof for ε0 is obtained by repeating the above argument using the property

that ε0Ti = −ε0t−1/2i and comparing coefficients in∑

w∈W0

τ∨wcw(Y )(−t−1/2i ) = ε0(−t−1/2

i ) = ε0Ti =∑w∈W0

τ∨wcw(Y )Ti

=∑w∈W0

τ∨wcw(Y )

(τ∨i −

t−1/2i − t1/2i

1− Y −α∨i

).

45

Corollary 3.16. [30, (5.7.8)], [8, (4.13)] For µ ∈ (h∗Z)+,

Pµ1 = t−1/2wµ Wµ(t) ·

∑v∈Wµ

∏a∨∈m−1

µ L(v−1,v−1µ )

t1/2a∨

1− qsh(−a∨)tht(−a∨)t−1a∨

1− qsh(−a∨)tht(−a∨)Evµ1.

and for µ ∈ (h∗Z)++,

Aµ1 =∑w∈W0

(−1)`(w)∏

a∨∈m−1µ L(w−1,w0)

t−1/2a∨

1− qsh(−a∨)tht(−a∨)ta∨

1− qsh(−a∨)tht(−a∨)Ewµ1,

Proof. Every element w in W0 has a unique factorization w = vu for v ∈ W µ and

u ∈ Wµ. Let b(a∨) = t1/2a∨

1− t−1a∨Y

−a∨

1− Y −a∨. Following Proposition 3.15,

Pµ1 = 10τ∨mµ1 =

∑w∈W0

τ∨w∏

a∨∈L(w−1,w−10 )

b(a∨)τ∨mµ1

=∑v∈Wµ

∑u∈Wµ

τ∨v τ∨u

∏a∨∈L((vu)−1,(vµu)−1)

b(a∨)

∏a∨∈L((vµu)−1,(vµwµ)−1)

b(a∨)

τ∨mµ1

=

∑v∈Wµ

τ∨v∏

a∨∈L((vu)−1,(vµu)−1)

b(ua∨)

∑u∈Wµ

τ∨u∏

a∨∈L(u−1,w−1µ )

b(a∨)

τ∨mµ1

=

∑v∈Wµ

τ∨v∏

a∨∈L(v−1,v−1µ )

b(a∨)

∑u∈Wµ

t−1/2wµu Tu

τ∨mµ1

= t−1/2wµ Wµ(t)

∑v∈Wµ

τ∨v∏

a∨∈L(v−1,v−1µ )

b(a∨)τ∨mµ1

= t−1/2wµ Wµ(t)

∑v∈Wµ

τ∨v τ∨mµ

∏a∨∈L(v−1,v−1

µ )

b(mµa∨)1

= t−1/2wµ Wµ(t)

∑v∈Wµ

∏a∨∈m−1

µ L(v−1,v−1µ )

t1/2a∨

1− qsh(−a∨)tht(−a∨)t−1a∨

1− qsh(−a∨)tht(−a∨)Evµ1.

We remark thatm−1µ v−1

µ = (vµxµv−1

µ )−1 = x−w0µ. If µ is a regular weight, thenW µ = W0,

so the proof for Aµ1 is more straightforward, and follows directly by applying ε0 to

τ∨mµ1.

46

Chapter 4

Littlewood-Richardson formulas

4.1 Littlewood-Richardson formulas

The following theorem expresses the product of a nonsymmetric Macdonald polynomial

and a symmetric Macdonald polynomial, in terms of nonsymmetric Macdonald polyno-

mial.

Theorem 4.1. Let µ ∈ h∗Z, λ ∈ (h∗Z)+, and fix a reduced expression for the minimal

length representative mµ = s∨ir · · · s∨i1

(π∨j )−1 of the coset xµW0. Then

EµPλ1 =∑w∈Wλ

∑h∈ΓC

2(~m−1µ ,(wmλ)−1)

ah(q, t) E$(h)1,

where ΓC2(~m−1

µ , (wmλ)−1) is the set of walks of type ~m−1

µ = (π∨j , s∨i1, . . . , s∨ir) beginning

in (wmλ)−1 for w ∈ W λ, contained in the dominant chamber C, and whose folds are

assigned the colours black or grey.

Define $(h) ∈ h∗Z by m$(h) = e(h)−1, where e(h) the ending alcove of h. The

47

coefficient ah(q, t) = bhnhfhgh, where

bh =∏

a∨∈L(i(h),m−1λ w0)

t1/2a∨

1− qsh(−a∨)tht(−a∨)t−1a∨

1− qsh(−a∨)tht(−a∨),

i(h) is the beginning alcove of h, and L(v, w) is defined in (2.18),

nh =∏

kth step of his a short crossing

(1− qsh(−h∨k )tht(−h∨k )t−1

ik

)(1− qsh(−h∨k )tht(−h∨k )

) (1− qsh(−h∨k )tht(−h∨k )tik

)(1− qsh(−h∨k )tht(−h∨k )

) ,

for short crossings defined in (2.63), and h∨k is such that the kth step of h

crosses or folds against Hh∨k , see (2.60),

fh =∏

kth step of ϑ(h)is a black fold

t−1/2

b∨k− t1/2b∨k

1− qsh(−b∨k )tht(−b∨k )

∏kth step of ϑ(h)

is a neg. black fold

qsh(−b∨k )tht(−b∨k ),

for ϑ(h) defined in (4.4), negative folds defined in (2.62),

and b∨k = s∨i1 · · · s∨ik−1

α∨ik , see (2.61),

gh = (−1)#long grey folds∏

kth step of his a grey fold

t−1/2ik− t1/2ik

1− qsh(−h∨k )tht(−h∨k )

∏k∈ψgr(h)

qsh(−h∨k )tht(−h∨k ),

ψgr(h) =

k∣∣∣∣ the kth step of h is a long grey fold and εk = −1

or the kth step of h is a short grey fold and εk = +1

,

for long and short folds defined in (2.63), and

εk =

+1, if the kth last step of ϑ(h) is negative,

−1, if the kth last step of ϑ(h) is positive.

48

Proof. The idea is to expand Eµ in terms of monomials Xν (Corollary 3.5), and then

expand XνPλ in terms of intertwiners (Proposition 3.15).

EµPλ1 = Eµ10τ∨mλ

1 =

∑p∈Γ(~mµ,1)

fpXe(p)

(∑w∈W0

bwτ∨wmλ

)1, (4.1)

with fp as in (3.6) and bw =∏

a∨∈m−1λ L(w−1,w0)

t1/2a∨

1− qsh(−a∨)tht(−a∨)t−1a∨

1− qsh(−a∨)tht(−a∨).

Identify e(p) with the sequence of alcoves given by the crossing steps of p to obtain

a (possibly non-reduced) expression for e(p). (For the walk p in Example 2.14, Xe(p) =

T1(T−11 )(T∨0 )−1(T−1

1 )(T∨0 )−1). Then by Corollary 3.7, a term in (4.1) is

fpbwXe(p)τ∨wmλ1 = fp

∑h∈ΓC(~e(p)−1,(wmλ)−1)

bhghnhτ∨e(h)−11 (4.2)

with gh and nh as defined in Corollary 3.7, and bh =∏

a∨∈L(i(h),m−1λ w0)

t1/2a∨

1− qsh(−a∨)tht(−a∨)t−1a∨

1− qsh(−a∨)tht(−a∨),

since p begins in (wmλ)−1. The condition that h must be contained in the dominant

chamber implies that its beginning alcove is one of (wmλ)−1 for w ∈ W λ, where W λ is

the set of minimal representatives of W0/Wλ.

The walks in ΓC(~e(p)−1, (wmλ)−1) can be viewed as those obtained by the translation

by (wmλ)−1 of the inverted walk of p (see Example 4.4), where the new folding steps are

distinguished by being coloured grey.

So let ΓC2(~m−1

µ , (wmλ)−1) denote the set of walks of type ~m−1

µ , beginning in (wmλ)−1

for w ∈ W0, contained in the dominant chamber C, and whose folds are assigned the

colours black or grey.

Given h ∈ ΓC2(~m−1

µ , (wmλ)−1), we can recover the walk p ∈ Γ(~mµ, 1) which generated

49

it, by letting

ζ(h) be the walk obtained from h by straightening all its grey folds, (4.3)

ϑ(h) be the walk obtained from ζ(h) by inverting its type, (4.4)

and translating it to begin at 1.

Then p = ϑ(h). (See Example 4.4.) Thus, following (4.1) and (4.2),

EλPµ1 =∑

p∈Γ(~mµ,1)

fp∑

h∈ΓC(~e(p)−1,(wmλ)−1)

bhnhghτ∨e(h)−11,

=∑w∈Wλ

∑h∈ΓC

2(~m−1µ ,(wmλ)−1)

fhbhnhgh E$(h)1,

where $(h) is the weight defined by e(h)−1 = m$(h).

Remark 4.2. By Corollary 3.16, the coefficient bh in Theorem 4.1 can also be expressed

as

bh = t−1/2wλ

Wλ(t)∏

a∨∈L(i(h),x−w0λ)

t1/2a∨

1− qsh(−a∨)tht(−a∨)t−1a∨

1− qsh(−a∨)tht(−a∨).

Remark 4.3.

1. Using interpolation Macdonald polynomials, Baratta obtained Type A Pieri formu-

las [1, Proposition 8, 10] for the expansion of Eµ(q, t)Pω1(0, 0) and Eµ(q, t)Pωn(0, 0)

in terms of nonsymmetric Macdonald polynomials. Similar formulas may also be

found in [21] (see note in the Introduction of [1]).

2. Haglund, Luoto, Mason, and van Willigenburg considered the Type A case at q =

t = 0, when Pλ(0, 0) is a Schur polynomial and Eλ(0, 0) is a Demazure character,

and obtained a formula for the expansion of Eµ(0, 0)Pλ(0, 0) in terms of Eµ(0, 0)

with positive coefficients [16, Theorem 6.1]. The coefficients∑

h:$(h)=γ a$(h)(0, 0)

count certain fillings of skew tableau-like diagrams called skyline diagrams.

50

Example 4.4. A Type sl2C example. The following walk p has type ~m−8ω = (s1s∨0 )4,

and is one of the 28 walks which appear in the expansion of E−8ω in terms of monomials.

•

Hα∨ H−α∨+2d H−α∨+4d H−α∨+6d H−α∨+8d

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

+−

1 x4ωs1

p :....................................................................................................................................................................................................................................................................... ................ .................................................................................................................... ................ .................................................................................................................... ................ .................................................................................................................... ................ .........................................................................................................................

............................................................................................................................................................. ................ .................................................................................................................... ................

In particular, the crossing steps of p give

Xe(p) = X4ωT1 = T1(T1)−1(T0)−1(T1)−1(T0)−1. (4.5)

Combined with the folding coefficients, the contribution of h to the expansion of E−8ω

in terms of monomials is

t1/2d(p)fpX

wt(p) = t1/2(t−1/2 − t1/2)

1− q7t

(t−1/2 − t1/2)q3t

1− q3t

(t−1/2 − t1/2)

1− q2tX4ω.

How the walk p generates some of the walks that appear in the expansion of E−8ωP2ω

in terms of nonsymmetric Macdonald polynomials is explained below. The translation

by x2ω of the inverse of p is the following walk h1 of type ~m−1−8ω = (s∨0 s1)4 beginning in

(s1m2ω)−1 is:

•

Hα∨ H−α∨+2d H−α∨+4d H−α∨+6d H−α∨+8d

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

...........................................................................................................

+−

1 x2ω

h1 :.................................................................................................................... ................ .........................................................................................................................

............................................................................................................................................................. ................ .................................................................................................................... ................ .................................................................................................................... ................ .................................................................................................................... ................ .......................................................................................................................................................................................................................................................................................

It is one of the walks which appear in the expansion of E−8ωP2ω in terms of nonsymmetric

Macdonald polynomials. It gives rise to the coefficients bh1 = 1,

nh1 =1− q6

1− q6t

1− q6t2

1− q6tgiven by the short crossings,

fh1 =(t−1/2 − t1/2)

1− q2t

(t−1/2 − t1/2)q3t

1− q3t

(t−1/2 − t1/2)

1− q7tgiven by the black folds,

51

gh1 = 1 because there are no grey folds, and $(h1) = 6ω (see Figure 2.5).

The following walk h2 ∈ ΓC2(~m−1

−8ω,m−12ω ) is another one of the walks appearing in the

expansion of E−8ωP2ω in terms of nonsymmetric Macdonald polynomials, obtained by

folding the 4th and 8th steps of h1, indicated in grey:

•

Hα∨ H−α∨+2d H−α∨+4d H−α∨+6d H−α∨+8d

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................

..................................................................................................................

..................................................................................................................

..................................................................................................................

..................................................................................................................

..................................................................................................................

..................................................................................................................

..................................................................................................................

..................................................................................................................

..................................................................................................................

+−

1 x2ω

h2:.................................................................................................................... ................ .........................................................................................................................

............................................................................................................................................................. ................ .........................................................................................................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................... ................ ...................................................................................................................................................

Note that

ζ(h2) = h1, and ϑ(h2) = p. (4.6)

By Equation (4.5)

Xe(ϑ(p)) = Xe(p) = (T∨i8)ε8(T∨i6)ε6(T∨i5)ε5(T∨i4)ε4(T∨i1)ε1 = T1(T1)−1(T0)−1(T1)−1(T0)−1,

so (ε8, ε6, ε5, ε4, ε1) = (+1,−1,−1,−1,−1). Both grey folds (the 4th and 8th) are long

folds, hence

ψgr(h2) =

k∣∣∣∣∣ the kth step of p is a long grey fold and εk = −1

or the kth step of p is a short grey fold and εk = +1

= {4}.

Therefore, the walk h2 gives rise to the coefficients bh2 = 1,

nh2 =1− q3

1− q3t

1− q3t2

1− q3t

1− q2

1− q2t

1− q2t2

1− q2tgiven by short the crossings,

fh2 =(t−1/2 − t1/2)

1− q2t

(t−1/2 − t1/2)q3t

1− q3t

(t−1/2 − t1/2)

1− q7tgiven by the black folds,

gh2 = (−1)2 (t−1/2 − t1/2)q4t

1− q4t

(t−1/2 − t1/2)

1− q2tgiven by the grey folds,

and $(h2) = 2ω. �

52

The following theorem expresses the product of two symmetric Macdonald polyno-

mials in terms of symmetric Macdonald polynomials.

Theorem 4.5. Let µ, λ ∈ (h∗Z)+, and fix a reduced expression for the minimal length

representative mµ = s∨ir · · · s∨i1

(π∨j )−1 of the coset xµW0. Then

PµPλ1 =∑w∈Wλ

∑h∈ΓC

2(~m−1µ ,m−1

λ w−1)

ch(q, t) P−w0wt(h)1,

where ΓC2(~m−1

µ ,m−1λ w−1) is the set of walks of type ~m−1

µ beginning in (wmλ)−1 for w ∈

W λ, contained in the dominant chamber C, and whose folds are assigned the colours

black or grey.

The coefficient ch(q, t) = ah(q, t)eh, where ah(q, t) is as in Theorem 4.1, and

eh =∏

a∨∈L(mwt(h),e(h))

t−1/2a∨

1− qsh(−a∨)tht(−a∨)ta∨

1− qsh(−a∨)tht(−a∨),

e(h) is the alcove where h ends, and L(v, w) is as defined in (2.18).

Proof. Since Pµ10 = 10Eµ10, then by Theorem 4.1,

PµPλ1 = 10EµPλ1 =∑w∈Wλ

∑h∈ΓC

2(~m−1µ ,(wmλ)−1)

a$(h)(q, t)10τ∨e(h)−11.

The weight $(h) is defined by e(h)−1 = m$(h), so let z ∈ W0 be of minimal length

such that z$(h)+ = $(h). By Proposition 3.12,

10τ∨e(h)−11 = 10τ

∨z τ∨m$(h)+

1 =

∏a∨∈L(mwt(h),e(h))

t−1/2a∨

1− qsh(−a∨)tht(−a∨)ta∨

1− qsh(−a∨)tht(−a∨)

P−w0wt(h)1,

since the ending alcove is e(h) = (zm$(h)+)−1, and $(h)+ = −w0wt(h) implies m$(h)+ =

m−w0wt(h) = m−1wt(h).

53

Remark 4.6. Let ¯ : K[X] → K[X] be the involution given by Xµ = X−µ, extended

linearly. By [8, p. 496], E−w0µ = w0Eµ, so applying the symmetrizing operator gives

P−w0µ = Pµ.

4.2 Specialization at q = 0

The symmetric Macdonald polynomials Pµ(0, t) are Hall-Littlewood polynomials, and in

this case, the product formulas in Theorem 4.1 and Theorem 4.5 require fewer walks to

compute. In fact, the walks in ΓC2(~m−1

µ , (wmλ)−1) which survive this specialization are

the precisely the those whose folds are positive and grey (from here on, these shall be

referred to as the ‘grey positively folded walks’). Theorem 4.5 essentially reduces to [37,

Theorem 1.3]. Also see [34, Theorem 4.9].

It is necessary to first consider Corollary 3.5, which is the expansion of Eµ into

monomials. In the case q = 0. Equation (3.6) becomes

fp =∏

k∈φ+(p)

t−1/2

b∨k− t1/2b∨k

1− qsh(−b∨k )tht(−b∨k ),

where φ+(p) = {k | the kth step of p is a positive fold}, so only the positively folded

walks in Γ(~mµ, 1) survive. If µ is a dominant weight, then the unique walk without folds

in Γ(~mµ, 1) is contained in the dominant chamber, and so consists of positive crossings

only. Thus if a walk in p ∈ Γ(~mµ, 1) has a fold, then it has at least one negative fold.

Therefore,

Eµ(0, t) = t1/2

v−1µXµ, if µ is dominant.

Likewise, by Corollary 3.14, the symmetric Macdonald polynomial Pµ(0, t) is a sum over

positively folded walks. It follows that for dominant weights λ and µ, the product of

54

Hall-Littlewood polynomials takes the form

Pµ(0, t)Pλ(0, t)1 = 10Xmµ10τ

∨mλ

(0, t)1,

and if h ∈ ΓC2(~m−1

µ , (wmλ)−1) is a walk with a black fold, then fh = 0. Thus any walk

with a black fold does not contribute to the product Pµ(0, t)Pλ(0, t).

Put another way, if µ is a dominant weight and h ∈ ΓC2(~m−1

µ , (wmλ)−1) does not

contain black folds, then the walk ϑ(h) (see (4.4)) is contained in the dominant chamber

and consists only of positive crossings. Thus

ψgr(h) = {k | the kth step of h is a long grey fold},

and there is a simplification of the coefficient gh in Theorem 4.5:

Case 1. If the kth step of h is a fold against an affine hyperplane Hh∨k which does not

contain the origin, then −h∨k = β∨ − jd, where j > 0, so qsh(−h∨k )tht(−h∨k ) = qjt〈ρ,β∨〉 = 0.

Case 2. If the kth step of h is a fold against a hyperplane Hh∨k which contains the

origin (and such a fold is necessarily long), then −h∨k = −α∨i for some i = 1, . . . , n, so

qsh(−h∨k )tht(−h∨k ) = q0t−〈ρ,α∨i 〉 = t−1.

So if h survives the specialization q = 0, then the only kinds of folds that h can have,

are short folds against affine hyperplanes, and long folds against hyperplanes containing

the origin:

gh =∏

kth step of his a short grey fold

t−1/2ik− t1/2ik

1− qsh(−h∨k )tht(−h∨k )

∏kth step of h

is a long grey fold

(−qsh(−h∨k )tht(−h∨k )

) (t−1/2ik− t1/2ik

)1− qsh(−h∨k )tht(−h∨k )

=∏

kth step of hfolds against aff. hyp.

(t−1/2ik− t1/2ik

) ∏kth step of h

folds against Hα∨i

t−1/2ik

.

Since the walks h are contained in the dominant chamber, then short folds against affine

hyperplanes and long folds against hyperplanes containing the origin are necessarily

55

positive. Therefore, the walks in ΓC2(~m−1

µ ,m−1λ w−1) which survive the specialization

q = 0 are the grey positively folded walks which are contained in the dominant chamber.

This explains why positive folds against the walls of the fundamental chamber versus

other affine walls contribute different coefficients in [34, Theorem 4.9] and [37, Theorem

1.3].

The remaining coefficients in Theorem 4.5 also simplify:

nh = 1, fh = 1, bh =∏

a∨∈L(m−1λ w0,i(p))

t1/2a∨ , eh =

∏a∨∈L(mwt(p),e(p))

t−1/2a∨ ,

and if in addition, λ and wt(p) are regular weights, and ti = t for all 0 ≤ i ≤ n, then

bh = t`(i(h))/2, eh = t−`(c(h))/2 = t`(d(h))/2−`(w0)/2, gh = (1− t)f(h)−f0(h)t−f(h)/2,

where f(h) is the number of folds, and f0(h) is the number of folds against a hyperplane

containing the origin. When λ and wt(h) are not regular, there is an extra factor that

depends on the length of the longest element of the stabilizer of the weight, but this is

due to the different normalization for Pµ chosen here.

Lastly, the formulas [34, Theorem 4.9] and [37, Theorem 1.3] use walks of type

~mµ while Theorem 4.5 uses walks of inverse type ~m−1µ . This difference is explained

by the automorphism −w0 of the lattice h∗Z, which preserves dominant weights. Let

¯ : K[X] → K[X] be the involution given by Xµ = X−µ, extended linearly. By [8, p.

496], E−w0µ = w0Eµ, and applying the symmetrizing operator gives P−w0µ = Pµ. So

−w0 : ΓC2

(~m−1µ , (wmλ)

−1)−→ ΓC

2

(~mµ,mλ(w0w

−1w0))

h 7→ −w0(h)

is a bijection such that ch(q, t) = c−w0(h)(q, t) and −w0wt(h) = wt(−w0(h)).

56

The last part of Example 5.4 is an illustration of the Littlewood-Richardson formulas

for Type A1 Macdonald polynomials at the specialization q = 0.

Remark 4.7. The Littlewood-Richardson rule for Hall-Littlewood polynomials Pλ(0, t)

in [37, Theorem 1.3] has a condition on the final direction d(h) of the walk h, while this

condition is absent in Theorem 4.5. This difference is due to the choice in the normal-

ization of Pλ(0, t), and is only up to scalar multiples by certain Poincare polynomials.

4.3 Pieri formulas

This section discusses the special cases of Theorem 4.1 and Theorem 4.5 when µ = ωr ∈

h∗Z is a minuscule weight (see Section 2.3).

The minimal coset representative mµ = π∨r ∈ Π∨ in this case, and Eωr1 = π∨r 1 =

XωrTv−1ωr

1 = t1/2vωrX

ωr1. Moreover, the walks appearing in Theorem 4.1 have type (π∨r )−1,

which is a “change in sheets”. Such walks do not have crossings or foldings, so the prod-

uct formula simplifies significantly. Combined with Corollary 3.16 (also see Remark 4.2),

then for λ ∈ (h∗Z)+,

EωrPλ1 = t−1/2wλ

Wλ(t)∑

h∈Γ((π∨r )−1,(Wλmλ)−1)

bh(q, t) E$(h)1, (4.7)

where Γ((π∨r )−1, (W λmλ)−1) is the set of walks of type (π∨r )−1 beginning in (wmλ)

−1 for

w ∈ W λ, $(h) is the weight defined by e(h)−1 = m$(h), and

bh(q, t) =∏

a∨∈L(i(h),x−w0λ)

t1/2a∨

1− qsh(−a∨)tht(−a∨)t−1a∨

1− qsh(−a∨)tht(−a∨). (4.8)

These conditions are enough to guarantee that the walks are contained in the dominant

chamber. Moreover, since i(h) = m−1λ w−1 for w ∈ W λ, and e(h) = i(h)(π∨r )−1 =

57

((π∨r )wmλ)−1, then m$(h) = e(h)−1 = (π∨r )wmλ implies $(h) = π∨r wλ = v−1

ωr wλ + ωr,

so (4.7) may also be written as

EωrPλ1 = t−1/2wλ

Wλ(t)∑w∈Wλ

∏a∨∈m−1

λ L(w−1,v−1λ )

t1/2a∨

1− qsh(−a∨)tht(−a∨)t−1a∨

1− qsh(−a∨)tht(−a∨)

Ev−1ωr wλ+ωr

1.

Likewise, Theorem 4.5 also simplifies when µ = ωr is a minuscule weight.

PωrPλ1 = t−1/2wλ

Wλ(t)∑

h∈Γ((π∨r )−1,(Wλmλ)−1)

bh(q, t)eh(q, t) P−w0wt(h)1, (4.9)

where bh(q, t) is as in (4.8), and

eh(q, t) =∏

a∨∈L(mwt(h),e(h))

t−1/2a∨

1− qsh(−a∨)tht(−a∨)ta∨

1− qsh(−a∨)tht(−a∨). (4.10)

Compression of the Pieri formula

The formula (4.9) is a sum over |W λ| walks, and many of the walks have the same

weight. By imposing a condition on the final direction of the walks and modifying the

coefficients appropriately, the formula can be compressed to contain the minimal number

of terms. Recall from Equation (2.19) that the element vωr ∈ W0 is the shortest such

that vωrωr = w0ωr. The maps

¯ : W−w0µ → Wµ : u 7→ u = v−1ωr uvωr , and ¯ : W−w0µ → W µ : v 7→ v = vvωr ,

are bijections.

Assuming λ is a dominant weight, then m−1λ = m−w0λ implies m−1

λ W0 = x−w0λW0.

Suppose v ∈ W−w0ωr and u ∈ W−w0ωr . A walk h of type (π∨r )−1 that begins in the alcove

x−w0λvu will end in the alcove x−w0λvu(π∨r )−1, and

e(h) = x−w0λvu(π∨r )−1 = x−w0λv(π∨r )−1u = x−w0λx−vvωrωr vu, (4.11)

58

so every walk h which begins in x−w0λvW−w0ωr has the same weight

wt(h) = −w0λ− vw0ωr = −w0(λ+ w0vw0ωr) = −w0(λ+ w0vωr).

In particular, these walks ends in xwt(h)vWωr . Thus (4.9) can be factorized as

PωrPλ1 = t−1/2wλ

Wλ(t)∑

v∈Wωr

∑h:e(h)∈Cv

bh(q, t)eh(q, t)

Pλ+w0vωr1, (4.12)

where Cv = xwt(h)vWωr ∩C. In the following, assume Cv = xwt(h)vWωr . This is equivalent

to the assumption that λ+ vωr is a regular weight, or λ− 2ρ is dominant.

Fix v ∈ W ωr and suppose h is a walk ending in xwt(h)vu ∈ Cv. By (4.11),

e(h)−1 = (mwt(h)vwt(h)vu)−1 = (vwt(h)vu)−1m−1wt(h) =

(vwt(h)vu

)−1m−w0wt(h),

so by (4.10), eh(q, t) is a product over the coroots

L(mwt(h), e(h)) = L(mwt(h), xwt(h)vwωr) t L(xwt(h)vwωr , xwt(h)vu). (4.13)

Also, by (4.8), bh(q, t) is a product over the coroots

L(i(h), x−w0λ) = L(x−w0λvu, x−w0λv) t L(x−w0λv, x−w0λ)

= L(xwt(h)vu, xwt(h)v) t L(x−w0λv, x−w0λ),(4.14)

so the decompositions (4.13) and (4.14) imply there is a common factor bheh in

∑h:e(h)∈Cv

bh(q, t)eh(q, t) = bheh∑

h:e(h)∈Cv

∏a∨∈xwt(h)vL(u,1)

b(a∨)∏

a∨∈xwt(h)vL(wωr ,u)

e(a∨)

, (4.15)

where

bh =∏

a∨∈x−w0λL(v,1)

b(a∨), eh =∏

a∨∈L(mwt(h),xwt(h)vwωr )

e(a∨),

and

b(a∨) = t1/2a∨

1− qsh(−a∨)tht(−a∨)t−1a∨

1− qsh(−a∨)tht(−a∨), e(a∨) = t

−1/2a∨

1− qsh(−a∨)tht(−a∨)ta∨

1− qsh(−a∨)tht(−a∨).

59

By Equation (2.16),

∑u∈Wωr

∏α∨∈L(1,u)

b(α∨)∏

α∨∈L(u,wωr )

e(α∨) =∑u∈Wωr

∏α∨∈R∨+uα∨∈R∨−

b(−uα∨)∏

α∨∈R∨+uα∨∈R∨+

e(uα∨)

=∑u∈Wωr

∏α∨∈R∨+uα∨∈R∨−

t1/2ua∨

1− t−1ua∨Y

ua∨

1− Y ua∨

∏α∨∈R∨+uα∨∈R∨+

t−1/2uα∨

1− tuα∨Y −uα∨

1− Y −uα∨1

= t−1/2wωr

∑u∈Wωr

∏β∨∈R∨+

1− tuα∨Y −uα∨

1− Y −uα∨1 = t−1/2

wωrWωr(t), (4.16)

where the last equality is [31, Corollary 2.6]. Define xwt(h)v : Q(t1/2)[Y ]1→ Q(q1/e, t1/2)[Y ]1

by xwt(h)vY λ∨1 = Y xwt(h)vλ∨1, extended linearly, so that in particular, xwt(h)vY 01 = 1.

Then by (4.16),

∑h:e(h)∈Cv

∏a∨∈x−w0λvL(u,1)

b(a∨)∏

a∨∈xwt(h)vL(wωr ,u)

e(a∨)

= xwt(h)v

∑u∈Wωr

∏α∨∈L(1,u)

b(α∨)∏

α∨∈L(u,wωr )

e(α∨)

= t−1/2wωr

Wωr(t). (4.17)

Therefore, if h ends in xwt(h)v (so that it has final direction d(h) = v ∈ W ωr), then

putting (4.12), (4.15) and (4.17) together and assuming λ− 2ρ is dominant,

PωrPλ = t−1/2wλ

Wλ(t) · t−1/2wωr

Wωr(t) ·∑h

bh(q, t)eh(q, t)P−w0wt(h), (4.18)

where

bh(q, t) =∏

a∨∈L(x−w0λd(h)v−1ωr ,x

−w0λ)

t1/2a∨

1− qsh(−a∨)tht(−a∨)t−1a∨

1− qsh(−a∨)tht(−a∨),

eh(q, t) =∏

a∨∈L(mwt(h),xwt(h)d(h)wωr )

t−1/2a∨

1− qsh(−a∨)tht(−a∨)ta∨

1− qsh(−a∨)tht(−a∨),

and the sum is over all walks h of type (π∨r )−1 beginning in (W λmλ)−1 with final direction

d(h) ∈ W ωr . See Example 5.15 for an illustration of this formula.

60

Remark 4.8. After renormalizing Pωr and Pλ by dividing by the appropriate Poincare

polynomials, Equation (4.18) is equivalent to Macdonald’s Pieri formula [27, (6.24)(iv)]

in terms of partitions. The case of Prω1Pλ is more involved, and is likely related to the

compression phenomenon described in [22].

61

Chapter 5

Examples

5.1 Type A1 examples

The roots and coroots for the complex simple Lie algebra sl2C are

R = {±α}, R∨ = {±α∨},

and the weight and coweight lattices are hZ = Zω∨, and h∗Z = Zω, where α = 2ω and

α∨ = 2ω∨. The Weyl group of this root system is the symmetric group on two symbols

W0 = S2 = 〈s1 | s21 = 1〉, where s1 is the reflection in the hyperplane Hα∨ = {0}.

The extended affine Weyl group W∨ is generated by the group W0 and π∨ = xωs1,

subject to the relations

(π∨)2 = 1, π∨s∨0 = s1π∨, (5.1)

where s∨0 = xαs1. Alternatively,

W∨ = {xkωw | k ∈ Z, w ∈ S2}. (5.2)

The following is the alcove picture for the extended affine Weyl group W∨, showing

the correspondence between the alcoves and the elements of W∨. The periodic orienta-

tion is indicated by + and − on either side of the hyperplanes. The two ways of indexing

the alcoves correspond to the two presentations (5.1) and (5.2) for W∨.

62

Figure 5.1.

•

H−α∨+2dHα∨Hα∨+2d H−α∨+d H−α∨+3dHα∨+d

Sheet 1 ...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...............................................................................

...............................................................................

...............................................................................

...............................................................................

...............................................................................

...............................................................................

...............................................................................

...............................................................................

+− +− +− +− +− +−1 s∨0 s∨0 s1 s∨0 s1s

∨0s1s1s

∨0s1s

∨0 s1

1 xαs1 xα x2αs1s1x−αx−αs1

Sheet π∨ ...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................

......................................................................................

......................................................................................

......................................................................................

......................................................................................

......................................................................................

......................................................................................

......................................................................................

+− +− +− +− +− +−π∨ π∨s1 π∨s1s

∨0 π∨s1s

∨0 s1π∨s∨0π∨s∨0 s1π∨s∨0 s1s

∨0

xωs1 xω x3ωs1 x3ωx−ωx−ωs1x−3ω

The double affine braid group B is generated by the groups

Π = 〈π | π2 = 1〉, B = 〈T0, T1〉, {qkXjω | k ∈ 12Z, j ∈ Z} ∼= h∗Z ⊕ 1

2Zδ,

subject to the relations

πT0π−1 = T1, T1X

ωT1 = X−ω, πXωπ−1 = q1/2X−ω,

and the double affine Hecke algebra H over the field K = Q(q1/2, t1/2) is the algebra

generated by the group algebra KB subject to the relations T 2i = (t1/2 − t−1/2)Ti + 1,

for i = 0, 1. Next, define

T∨1 = T1, (T∨0 )−1 = XαT1, Y ω∨ = πT1, π∨ = XωT1.

The duality theorem 2.6 can be directly verified.

Proposition 5.2. The double affine braid group B is generated by the groups

Π∨ = 〈π∨ | (π∨)2 = 1〉, B∨ = 〈T∨0 , T1〉, {qkY jω∨ | k ∈ 12Z, j ∈ Z} ∼= hZ ⊕ 1

2Zδ,

subject to the relations

π∨T∨0 (π∨)−1 = T1, T−11 Y ω∨T−1

1 = Y −ω∨, π∨Y ω∨(π∨)−1 = q−1/2Y −ω

∨.

63

Proof. Directly calculate

(π∨)2 = XωT1XωT1 = XωX−ω = 1,

Y ω∨Y ω∨ = πT1πT1 = T0T1 = Y α∨ ,

π∨T∨0 (π∨)−1 = (XωT1)(T−11 X−α)(π∨)−1 = X−ω(T−1

1 X−ω) = X−ω(XωT1) = T1,

T−11 Y ω∨T−1

1 = T−11 πT1T

−11 = T−1

1 π−1 = Y −ω∨,

π∨Y ω∨(π∨)−1 = π∨(πT1)(T−11 X−ω) = π∨q−1/2Xωπ = q−1/2XωT1X

ωπ−1 = q−1/2Y −ω∨.

Remark 5.3. The relation T0 = πT1π implies that

T0X−ωT0 = (πT1π)X−ω(πT1π) = πT1q

−1/2XωT1π = q−1/2πX−ωπ = q−1Xω,

and similarly, the relation T∨0 = π∨T1π∨ implies that

(T∨0 )−1Y −ω∨(T∨0 )−1 = π∨T−1

1 q1/2Y ω∨T−11 π∨ = q1/2π∨Y −ω

∨π∨ = qY ω∨ .

With Y −α∨0 = qY α∨ = qT0T1, and τ∨0 = π∨τ∨1 π

∨, the intertwiners are

π∨ = XωT1,

τ∨i = T∨i +t−1/2 − t1/2

1− Y −α∨i= (T∨i )−1 +

(t−1/2 − t1/2)Y −α∨i

1− Y −α∨i, for i = 0, 1,

For µ ∈ h∗Z, the minimal (left) coset representatives are

mkω =

xkωs1 = π∨(s1π

∨)k−1, k > 0,

xkω = (s1π∨)k, k ≤ 0.

So the nonsymmetric Macdonald polynomials are given by E01 = 1, and

Ekω1 = π∨(τ∨1 π∨)k−11, E−kω1 = τ∨1 Ekω1, for k ≥ 1.

64

The first few nonsymmetric Macdonald polynomials are

Eω1 = π∨1 = t1/2Xω1,

E−ω1 = τ∨1 π∨1 = X−ω1 +

1− t1− qt

Xω1,

E2ω1 = π∨τ∨1 π∨1 = t1/2X2ω1 + t1/2q

1− t1− qt

1,

E−2ω1 = τ∨1 π∨τ∨1 π

∨1 = X−2ω1 +1− t1− qt

1 +1− t

1− q2tX2ω1 + q

1− t1− q2t

1− t1− qt

1.

Example 5.4. Littlewood-Richardson formulas. In the following example, we use

the alcove walk formulas to calculate E3ω, E3ωPkω and P3ωPkω for k ≥ 3. We also look

at various specializations of the parameters q and t.

The minimal coset representative of x3ωW0 is m3ω = s∨0 s1π∨, so E3ω1 = τ∨0 τ

∨1 π∨1,

and by Corollary 3.5,

E3ω = X3ωt1/2 +(t−1/2 − t1/2)qt

1− qtXω +

(t−1/2 − t1/2)

1− qt(t−1/2 − t1/2)q2t

1− q2tXωt1/2

+(t−1/2 − t1/2)q2t

1− q2tX−ω,

(5.3)

where the four terms arise from the following four walks:

..................................................................................................................................................................................................................................................................................

.......

.......

.......

.......

.....

.......

.......

.......

.......

.....

.......

.......

.......

.......

.....

............................................................................................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.

.......

.......

.......

.......

.......

.......

.

.......

.......

.......

.......

.......

.......

.

•p1

x3ωs1

...................................................................... ................ ...................................................................... ................ ........

..................................................................................................................................................................................................................................................................................

.......

.......

.......

.......

.....

.......

.......

.......

.......

.....

.......

.......

.......

.......

.....

............................................................................................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.

.......

.......

.......

.......

.......

.......

.

.......

.......

.......

.......

.......

.......

.

•p2

xω

...................................................................... ................ ....................................................................................................

...

..................................................................................................................................................................................................................................................

.......

.......

.......

.......

.....

.......

.......

.......

.......

.....

.......

.......

.......

.......

.....

.......

.......

.......

.......

.....

............................................................................................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.

.......

.......

.......

.......

.......

.......

.

.......

.......

.......

.......

.......

.......

.

•p3

xωs1

.................................................................................................................................................................................. ................ ...

... ..................................................................................................................................................................................................................................................................................

.......

.......

.......

.......

.....

.......

.......

.......

.......

.....

.......

.......

.......

.......

.....

............................................................................................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.

.......

.......

.......

.......

.......

.......

.

.......

.......

.......

.......

.......

.......

.

•p4

x−ω

..........................................................................................................................................................................................

...

.

Let µ = 3ω and λ = kω. The walks in ΓC2(~m−1

µ , (wmλ)−1) for w = 1 or s1 have type

~m−1µ = (π∨, 1, 0) and begin in the alcoves m−1

λ = xkωs1 or (s1mλ)−1 = xkω.

There are eighteen walks in ΓC2(~m−1

µ , (wmλ)−1). The first eight walks are generated

by inverting the walk p1, translating it to begin at the two possible starting points xkω

or xkωs1, and adding new folds. The weights $(h) are indicated above each walk h in

the following pictures.

65

Generated by p1, these walks have fh = 1:

h bh eh nh gh

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

(k + 3)ω

....................................................................................................................

...

...

. 1 1 1 1

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

−(k + 1)ω

.............................................................................................................................

...

...

. 1 t−1/21− qk+1t2

1− qk+1t1 − (t−1/2 − t1/2)qk+2t

1− qk+2t

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

−(k − 1)ω

.............................................................................................................................

...

...

. 1 t−1/21− qk−1t2

1− qk−1t1− qk

1− qkt1− qkt2

1− qkt− (t−1/2 − t1/2)qk+1t

1− qk+1t

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

(k + 1)ω

..................................................................................................................................

...

... 1 1 1 − (t−1/2 − t1/2)qk+1t

1− qk+1t

t−1/2 − t1/2

1− qkt

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

−(k − 3)ω

....................................................................................................................

...

...

. t1/21− qk

1− qktt−1/2

1− qk−3t2

1− qk−3t

1− qk−1

1− qk−1t1− qk−1t2

1− qk−1t

· 1− qk−2

1− qk−2t1− qk−2t2

1− qk−2t

1

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

(k − 1)ω

.............................................................................................................................

...

...

. t1/21− qk

1− qkt1

1− qk−1

1− qk−1t1− qk−1t2

1− qk−1tt−1/2 − t1/2

1− qk−2t

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

(k + 1)ω

.............................................................................................................................

...

...

. t1/21− qk

1− qkt1 1

t−1/2 − t1/2

1− qk−1t

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

−(k − 3)ω

..................................................................................................................................

...

... t1/21− qk

1− qktt−1/2

1− qk−1t2

1− qk−1t1 − t

−1/2 − t1/2

1− qk−1t(t−1/2 − t1/2)qkt

1− qkt

66

Generated by p2, these walks have fh =(t−1/2 − t1/2)qt

1− qt:

h bh eh nh gh

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

−(k − 1)ω

.............................................................................................................................

...

...

. 1 t−1/21− qk−1t2

1− qk−1t1− qk

1− qkt1− qkt2

1− qkt1

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

(k + 1)ω

..................................................................................................................................

...

... 1 1 1t−1/2 − t1/2

1− qkt

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

(k + 1)ω

.............................................................................................................................

...

...

. t1/21− qk

1− qkt1 1 1

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

−(k − 1)ω

..................................................................................................................................

...

... t1/21− qk

1− qktt−1/2

1− qk−1t2

1− qk−1t1 − (t−1/2 − t1/2)qkt

1− qkt

Generated by p3, these walks have fh =(t−1/2 − t1/2)q2t

1− q2t

(t−1/2 − t1/2)

1− qt:

h bh eh nh gh

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

(k + 1)ω

..................................................................................................................................

...

... 1 1 1 1

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

−(k − 1)ω

..................................................................................................................................

...

... t1/21− qk

1− qktt−1/2

1− qk−1t2

1− qk−1t1 1

67

Generated by p4, these walks have fh =(t−1/2 − t1/2)q2t

1− q2t:

h bh eh nh gh

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

−(k + 1)ω

.............................................................................................................................

...

...

. 1 t−1/21− qk+1t2

1− qk+1t1 1

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

(k + 1)ω

..................................................................................................................................

...

... 1 1 1 − t−1/2 − t1/2

1− qk+1t

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

(k − 1)ω

.............................................................................................................................

...

...

. t1/21− qk

1− qkt1

1− qk−1

1− qk−1t1− qk−1t2

1− qk−1t1

......................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

.......

.......

....

kω

−(k − 1)ω

..................................................................................................................................

...

... t1/21− qk

1− qktt−1/2

1− qk−1t2

1− qk−1t1

(t−1/2 − t1/2)qk−1t

1− qk−1t

Thus E3ωPkω is a linear combination of E(k+3)ω, E−(k+1)ω, E(k+1)ω, E−(k−1)ω, E(k−1)ω,

E−(k−3)ω, and after simplification,

E3ωPkω = E(k+3)ω + q2 1− t1− q2t

1− qk

1− qk+2tt1/2E−(k+1)ω

+1− t1− q

1− q2

1− q2t

1− qk

1− qk−1t

1− qk+1t2

1− qk+1tE(k+1)ω

+ q1− t1− q

1− q2

1− q2t

1− qk−1

1− qk−1t

1− qk

1− qkt1− qkt2

1− qk+1tt1/2E−(k−1)ω

+1− t

1− q2t

1− qk−1

1− qk−2t

1− qk

1− qk−1t

1− qk−1t2

1− qk−1t

1− qkt2

1− qktE(k−1)ω

+1− qk−2

1− qk−2t

1− qk−2t2

1− qk−2t

1− qk−1

1− qk−1t

1− qk−1t2

1− qk−1t

1− qk

1− qktt1/2E−(k−3)ω.

68

Applying the symmetrizing operator 10 to the above equation gives the expression for

P3ωPkω in terms of P which are indexed by non-dominant weights. In order to get an ex-

pression in terms of P which are indexed by dominant weights only, use Proposition 3.12

P−$(h) = t−1/2 1− qjt2

1− qjtP$(h) = ehP−w0wt(h)

for $(h) = jω and j a positive integer. Thus, after simplification,

P3ωPkω = P(k+3)ω +1− t1− q

1− q3

1− q2t

1− qk

1− qk−1t

1− qk+1t2

1− qk+2tP(k+1)ω

+1− t1− q

1− q3

1− q2t

1− qk−1

1− qk−2t

1− qk

1− qk−1t

1− qk−1t2

1− qkt1− qkt2

1− qk+1tP(k−1)ω

+1− qk−2

1− qk−2t

1− qk−2t2

1− qk−2t

1− qk−1

1− qk−1t

1− qk−1t2

1− qk−1t

1− qk

1− qkt1− qk−3t2

1− qk−3tP(k−3)ω.

Consider the case q = 0 where the symmetric Macdonald polynomials are Hall-

Littlewood polynomials. Following the discussion in Section 4.2, the walks giving a

nonzero contribution to the sum are those whose only folds are positive and gray. The

expression

P3ω(0, t)Pkω(0, t) = P(k+3)ω(0, t) + (1− t)P(k+1)ω(0, t) + (1− t)P(k−1)ω(0, t) +P(k−3)ω(0, t),

is given by four positively folded walks, and coincides with the Littlewood-Richardson

formulas [37, Theorem 1.3] and [34, Theorem 4.9] for Hall-Littlewood polynomials. We

also mention

E3ω(0, t)Pkω(0, t) = E(k+3)ω(0, t) + (1− t)E(k+1)ω(0, t)

+ (1− t)E(k−1)ω(0, t) + t1/2E−(k−3)ω(0, t).

In the case q = t = 0, the symmetric Macdonald polynomials are Schur polynomials

Pµ(0, 0) = sµ, and the nonsymmetric Macdonald polynomials are Demazure characters

69

Eµ(0, 0) = Aµ. The dominant weights kω correspond to the partitions (k, 0), and the

weights −kω correspond to the compositions (0, k) for k ≥ 0. The four positively folded

walks correspond to the four Littlewood-Richardson tableaux that give the classical

Littlewood-Richardson formula for Schur functions:

P3ω(0, 0)Pkω(0, 0) = P(k+3)ω(0, 0) + P(k+1)ω(0, 0) + P(k−1)ω(0, 0) + P(k−3)ω(0, 0).

By normalizing the nonsymmetric polynomials Eµ so that the coefficient of the monomial

Xµ in Eµ is 1, (see the paragraph following (3.1)), then each term in

E3ω(0, 0)Pkω(0, 0) = E(k+3)ω(0, 0) + E(k+1)ω(0, 0) + E(k−1)ω(0, 0) + E−(k−3)ω(0, 0),

corresponds to a skyline filling in the formula [16, Theorem 6.1]. �

Example 5.5. Pieri formulas. The weight ω is minuscule in the sl2 root system. The

two walks which give the expressions

EωPkω = E(k+1)ω +1− qk

1− qktt1/2E−(k−1)ω,

PωPkω = P(k+1)ω +1− qk

1− qkt1− qk−1t2

1− qk−1tP(k−1)ω,

are walks of type ~m−1ω = π∨ corresponding to ‘changes in sheets’, and begin in the alcoves

m−1kω or m−1

kωs1. The following picture depicts these walks:

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•

H−α∨+kd

m−1kω m−1kωs1

m−1kωπ∨ m−1kωs1π

∨

. . . . .

. . . . .

...

...

...

.

...

...

...

.

For each walk h, only the statistics bh and eh are necessary for these calculations. The

statistic bh records the hyperplanes separating the beginning alcove of h and m−1kωs1 =

xkω, and eh records the hyperplanes separating the ending alcove of h and mwt(h). �

70

Example 5.6. For k ≥ 1,

Pkω = E−kω +1− qk

1− qktt1/2Ekω, (5.4)

Akω = −E−kω +1− qkt2

1− qktt−1/2Ekω, (5.5)

P−kω = t−1/2 1− qkt2

1− qktPkω, (5.6)

A−kω = −t1/2 1− qk

1− qktAkω. (5.7)

The calculation (5.4) can be regarded as an example of Corollary 3.16, or as a special

case of Theorem 4.1 with µ = 0 (so that Eµ = 1), which expresses Pkω as a linear

combination of Ekω and E−kω. The two walks which give the expression

Pkω = E−kω +1− qk

1− qktt1/2Ekω, k ≥ 1,

have trivial type ~m−10 = 1, and so they may be identified with the alcoves in the coset

m−1kωW0 = {m−1

kω ,m−1kωs1}:

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.

H−α∨+kd

• • •m−1kω m−1kωs1

. . . . .

For each walk h, only the statistic bh is needed for this calculation, and it records the

hyperplanes separating the alcoves i(h) and m−1kωs1 = xkω. The calculation (5.5) is

another example of Corollary 3.16, which is computed by the same two walks in the

above picture, using a slightly different statistic.

Equations (5.6) and (5.7) are examples of Proposition 3.12. �

71

Example 5.7. The following calculations are an application of Theorem 3.6. For k ≥ 1,

XωEkω = t−1/2EωEkω = E(k+1)ω −qk(1− t)1− qkt

t1/2E−(k−1)ω, (5.8)

X−ωE−kω = E−ωE−kω = E−(k+1)ω −1− t

1− qk+1tt−1/2E(k+1)ω, (5.9)

X−ωEkω = E−ωEkω =1− qk−1

1− qk−1t

1− qk−1t2

1− qk−1tE(k−1)ω +

qk−1(1− t)1− qk−1t

t1/2E−(k−1)ω, (5.10)

XωE−kω = t−1/2EωE−kω =1− qk

1− qkt1− qkt2

1− qktE−(k−1)ω +

1− t1− qkt

t−1/2E(k+1)ω. (5.11)

Note Xω = (T∨0 )−1π∨. Since z = xω = s∨0π∨ ∈ W∨, Equation (5.8) is a sum over the two

walks of type ~z−1 = (π∨, 0) beginning in the alcove m−1kω . The weight $(h) is indicated

above each walk:

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.

•

H−α∨+kd

m−1kω

(k + 1)ω

..............................................................................................

...

...

...

.

. . .

. . .

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.

•

H−α∨+kd

m−1kω

−(k − 1)ω

.........................................................................................................

...

...

...

.

. . .

. . .

Similarly, since X−ω = T1π∨, z = x−ω = s1π

∨ ∈ W∨, so Equation (5.9) is a sum over

the two walks of type ~z−1 = (π∨, 1) beginning in the alcove m−1−kω = m−1

kωs1. The weight

$(h) is indicated above each walk:

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.

•

H−α∨+(k+1)d

m−1kωs1

−(k + 1)ω

..............................................................................................

...

...

...

.

. . .

. . .

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.

•

H−α∨+(k+1)d

m−1kωs1

(k + 1)ω

.........................................................................................................

...

...

...

.

. . .

. . .

Equation (5.10) can be obtained by considering the two walks of type (π∨, 1) beginning

in m−1kω and Equation (5.11) can be obtained by considering the two walks of type (π∨, 0)

beginning in m−1kωs1. �

72

5.2 Type A2 examples

Let {ε1, ε2, ε3} be an orthonormal basis for R3, and let {ε∨1 , ε∨2 , ε∨3 } be its dual basis,

where 〈εi, ε∨j 〉 = δij. The simple roots and simple coroots of the complex simple Lie

algebra sl3C are

α1 = ε1 − ε2, α2 = ε2 − ε3, ϕ = ε1 − ε3,

α∨1 = ε∨1 − ε∨2 , α∨2 = ε∨2 − ε∨3 , ϕ∨ = ε∨1 − ε∨3 ,

so that the root and coroot lattices are Q = Zα1 + Zα2 and Q = Zα∨1 + Zα∨2 . The

fundamental weights and fundamental coweights are

ω1 = ε1 − 13(ε1 + ε2 + ε3), ω2 = ε1 + ε2 − 2

3(ε1 + ε2 + ε3),

ω∨1 = ε∨1 − 13(ε∨1 + ε∨2 + ε∨3 ), ω∨2 = ε∨1 + ε∨2 − 2

3(ε∨1 + ε∨2 + ε∨3 ),

and the weight and coweight lattices are h∗Z = Zω1 ⊕ Zω2 and hZ = Zω∨1 ⊕ Zω∨2 . The

group Π∨ ∼= h∗Z/Q is the cyclic group of order 3. Note that 〈h∗Z, hZ〉 ⊆ 13Z.

The Weyl group of this root system is the symmetric group on three symbols

W0 = S3 =⟨s1, s2, | s1s2s1 = s2s1s2 = sϕ, s

2i = 1 for i = 1, 2

⟩,

and the extended affine Weyl group W∨ is generated by the group W0 and the element

π∨ = xω1s1s2, subject to the relations

(π∨)3 = 1, π∨s∨0 = s1π∨, π∨s1 = s2π

∨, s0s1s0 = s1s0s1, s0s2s0 = s2s0s2, (5.12)

where s∨0 = xϕsϕ and (π∨)2 = xω2s2s1. Alternatively,

W∨ = {xk1ω1+k2ω2w | k1, k2 ∈ Z, w ∈ S3}. (5.13)

73

Figure 5.8 is the alcove picture for the extended affine Weyl group W∨, showing the

correspondence between the alcoves and the elements of W∨. The periodic orientation

is indicated by + and − on either side of the hyperplanes. Since the slnC root system is

self-dual, the dual alcove picture is identical.

The double affine braid group B is generated by the groups

Π = 〈π | π3 = 1〉,

B = 〈T0, T1, T2 | TiTjTi = TjTiTj for j = i+ 1 mod 3〉,

{qkXk1ω1+k2ω2 | k ∈ 13Z, k1, k2 ∈ Z} ∼= h∗Z ⊕ 1

3Zδ,

subject to the relations

πT0π−1 = T1, πT1π

−1 = T2, πT2π−1 = T0,

T1Xω1T1 = X−ω1+ω2 , T1X

ω2 = Xω2T1, T2Xω1 = Xω1T2, T2X

ω2T2 = Xω1−ω2 ,

πXω1π−1 = q1/3X−ω1+ω2 , πXω2π−1 = q2/3X−ω1 ,

πX−ω1π−1 = q−2/3Xω1 , πX−ω2π−1 = q−1/3Xω1−ω2 ,

and the double affine Hecke algebra H over the field K = Q(q1/3, t1/2) is the algebra

generated by the group algebra KB subject to the relations

T 2i = (t1/2 − t−1/2)Ti + 1, for i = 0, 1, 2.

74

Figure 5.8.

......................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................Hα∨2

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Hα∨1

...................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................... Hϕ∨

..................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................

+ −+−

+−Sheet (π∨)2 •

π∨2

π∨2s2

π∨2s1

π∨2s∨0

xω2

xω1−ω2

x−ω1

......................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................Hα∨2

...................................................................................................................................................................................................................................................................................................................................................................................

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Hα∨1

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............................................................................................................................................................................................................................................................................................................................................................................................................................................................................... Hϕ∨

..................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................

+ −+−

+−Sheet π∨ •

π∨

π∨s1

π∨s2

π∨s∨0

xω1

x−ω1+ω2

x−ω2

......................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................Hα∨2

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Hα∨1

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............................................................................................................................................................................................................................................................................................................................................................................................................................................................................... Hϕ∨

..................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................

+ −+−

+−Sheet 1 •

1s1 s2

s1s2 s2s1

sϕ

s∨0xα1xα2

x−α1x−α2

x−ϕ

75

Define

T∨1 = T1, T∨2 = T2, (T∨0 )−1 = XϕTsϕ ,

Y −α∨1 = T−1

1 T−12 T−1

0 T2, Y −α∨2 = T−1

2 T−11 T−1

0 T1, Y −α∨0 = qT0Tsϕ .

The intertwiners are

π∨ = Xω1T1T2,

(π∨)2 = Xω2T2T1,

τ∨i = T∨i +t−1/2 − t1/2

1− Y −α∨i= (T∨i )−1 +

(t−1/2 − t1/2)Y −α∨i

1− Y −α∨i, for i = 0, 1, 2,

where (π∨)3 = 1, π∨τ∨i (π∨)−1 = τ∨j and τ∨i τ∨j τ∨i = τ∨j τ

∨i τ∨j for j = i+ 1 mod 3.

Example 5.9. Using Theorem 3.4, we calculate the expansion of E−α21 in the monomial

basis. The minimal coset representative is m−α2 = s2s1s∨0 , and the eight walks of type

(s2, s1, s∨0 ) beginning in the fundamental alcove are:

76

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•

X−α2T1

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•

X0T2T1t−1/2 − t1/2

1− Y ϕ∨−d

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•

Xα1T1T2t−1/2 − t1/2

1− Y α∨2−d

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•

X0T2(t−1/2 − t1/2)Y ϕ

∨−d

1− Y ϕ∨−dt−1/2 − t1/2

1− Y α∨2 −d

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•

Xα2T2T1t−1/2 − t1/2

1− Y ϕ∨−2d

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•

X0T1t−1/2 − t1/2

1− Y ϕ∨−2d

(t−1/2 − t1/2)Y ϕ∨−d

1− Y ϕ∨−d

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•

XϕT1T2T1t−1/2 − t1/2

1− Y ϕ∨−2dt−1/2 − t1/2

1− Y α∨2−d

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•

X0 t−1/2 − t1/2

1− Y ϕ∨−2dt−1/2 − t1/2

1− Y α∨2−d(t−1/2 − t1/2)Y ϕ

∨−d

1− Y ϕ∨−d

...............................................................................................................................................................................................

77

Since Y ϕ∨−d1 = qt21, Y α∨2−d1 = qt1, and Y ϕ∨−2d1 = q2t21, then in the polynomial

representation,

t−1/2E−α21 = τ∨2 τ∨1 τ∨0 1

= X−α21 +Xα11− t1− qt

1 +Xα21− t

1− q2t21 +Xϕ 1− t

1− q2t21− t1− qt

1

+X0

(1− t

1− qt2+

1− t1− qt2

1− t1− qt

qt+1− t

1− q2t21− t

1− qt2qt+

1− t1− q2t2

1− t1− qt

1− t1− qt2

q

)1.

�

Example 5.10. We calculate Eϕ1 and its symmetric version Pϕ1 in the monomial basis.

The minimal coset representative is mϕ = s∨0 . Two walks give the computation for

Eϕ1 = τ∨0 1 = t3/2(Xϕ +

(1− t)q1− qt2

X0

)1

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•

XϕTsϕ

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•

(t−1/2 − t1/2)Y −α∨0

1− Y −α∨0

...............

.....................................................................

The following shows the symmetrization of the previous two paths for Eϕ1, giving

twelve paths for the computation of

Pϕ1 = 10τ∨0 1 = Mϕ1 + (2 + t+ q + 2qt)

1− t1− qt2

1

78

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•

Mϕ1 =∑λ∈W0ϕ

Xλ1

...............

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•

(2 + t+ q + 2qt)1− t

1− qt21

...............

......................................................... .......................................................................

..............................................................................................................................................

........................................................................

.......................................................................

The six folded walks may be grouped into two sets, (one set consists of the two leftmost

folded walks, the other set consists of the remaining four folded walks) to give the

factorization

Pϕ1 = Mϕ1 + ((1 + qt) + (1 + t+ q + qt))1− t

1− qt21

= Mϕ1 +

(1− q2t2

1− qt+

1− q2

1− q1− t2

1− t

)1− t

1− qt21.

Each of these sets contains one positively folded walk with maximal dimension, and such

walks correspond to LS-galleries. See [13, Definition 14], [37, Definition 4.7], and [33,

(2.4)] . �

Example 5.11. Using Theorem 3.6, we calculate the expansion of X−α21 in the basis

of nonsymmetric Macdonald polynomials. Since X−α1 = T2T1T∨0 T−11 , there are six-

teen walks of type (s1, s∨0 , s1, s2) beginning in the fundamental alcove, but only five are

contained in the dominant chamber. In particular, the first step must be folded:

79

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•

−τ∨m−α2

(t−1/2 − t1/2)Y −α∨1

1− Y −α∨1

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•

τ∨mα2

(t−1/2 − t1/2)Y −α∨1

1− Y −α∨1t−1/2 − t1/2

1− Y ϕ∨−2d

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•

τ∨mα1

(t−1/2 − t1/2)Y −α∨1

1− Y −α∨1t−1/2 − t1/2

1− Y α∨2 −d

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•

−τ∨mϕ(t−1/2 − t1/2)Y −α

∨1

1− Y −α∨1t−1/2 − t1/2

1− Y α∨2 −dt−1/2 − t1/2

1− Y α∨1 −d

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•

(t−1/2 − t1/2)Y −α∨1

1− Y −α∨1t−1/2 − t1/2

1− Y −α∨2t−1/2 − t1/2

1− Y −α∨1t−1/2 − t1/2

1− Y ϕ∨−d

...................................

...............................

......................................................................................................................................................................

In the polynomial representation,

X−α21 = T2T1T∨0 T−11 1

= t−1/2

(E−α21−

t−1/2 − t1/2

1− q2t2Eα21−

t−1/2 − t1/2

1− qtEα11

+t−1/2 − t1/2

1− qtt−1/2 − t1/2

1− qtEϕ1− t

t−1/2 − t1/2

1− qt2E0

)1.

�

Example 5.12. Using Corollary 3.16, we calculate the expansion of P3ω11 in the non-

symmetric Macdonald polynomial basis. The stabilizer of µ = 3ω1 is Wµ = {1, s2}, and

W µ = {1, s1, s2s1}, so v3ω1 = s2s1. The minimal coset representative is m3ω1 = s∨0 s2s1s∨0 .

The hyperplanes separating the alcoves m−13ω1

and m−13ω1v−1

3ω1are indicated below by dotted

80

lines.

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...................................

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•1

m−13ω1

m−13ω1

s1s2

H−ϕ∨+3d

Hα∨1

H−α∨2 +3d

P3ω11 = (t1/2 + t−1/2)

(τ∨s2s1(3ω1) + τ∨s1(3ω1)

t1/2 − t−1/2Y α∨2−3d

1− Y α∨2−3d

+ τ∨3ω1

t1/2 − t−1/2Y α∨2−3d

1− Y α∨2−3d

t1/2 − t−1/2Y ϕ∨−3d

1− Y ϕ∨−3d

)1

= (t1/2 + t−1/2)

(E−3ω2 +

1− q3

1− q3tt1/2E−3ω1+3ω2 +

1− q3

1− q3t

1− q3t

1− q3t2tE3ω1

)1.

�

Example 5.13. Using Proposition 3.12, we calculate the scalar multiple difference be-

tween Ps2s13ω11 and P3ω11. The minimal coset representative is m3ω1 = s∨0 s2s1s∨0 . The

two hyperplanes separating the alcoves m−1ω1

and m−1ω1

(s2s1)−1 are indicated below by

dotted lines.

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•1

m−13ω1

m−13ω1

s1s2

H−ϕ∨+3d

Hα∨1

H−α∨2 +3d

81

Ps2s13ω11 = 10τ∨3ω1

t−1/2 − t1/2Y α∨2−3d

1− Y α∨2−3d

t−1/2 − t1/2Y ϕ∨−3d

1− Y ϕ∨−3d1 =

1− q3t2

1− q3t

1− q3t3

1− q3t2t−1P3ω11.

�

Example 5.14. Using Theorem 4.5, we calculate P 2ϕ1. The minimal coset representative

is mϕ = s∨0 , and sixteen of the eighteen walks of type s∨0 beginning in (ws∨0 )−1 for w ∈ S3

are contained in the dominant chamber.

P2ϕ

t1/21− q1− qt

P3ω1t1/2

1− qt1− qt

P3ω2

t3/21− q1− qt

1− q1− qt

1− q2t1− q2t2

1− qt1− qt2

1− qt3

1− qt2P0

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six walks with weight ϕ

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(t−1/2 − t1/2)qt2

1− qt2t−3/2W0(t)Pϕ

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82

The six walks with grey folds and weight ϕ are

− (t−1/2 − t1/2)q3t2

1− q3t21− qt2

1− qt1− q2t3

1− q2t21− qt2

1− qtt−3/2Pϕ

− 2(t−1/2 − t1/2)q2t

1− q2t

1− q1− qt

1− qt2

1− qt1− q2t3

1− q2t2t−1/2Pϕ

+ 21− q1− qt

1− q2t

1− q2t2t−1/2 − t1/2

1− t1− qt2

1− qtt1/2Pϕ

+1− q1− qt

1− q1− qt

1− q2t

1− q2t2t−1/2 − t1/2

1− qt2t3/2Pϕ.

�

Example 5.15. Using Theorem 4.5, we first calculate Pω1P2ω1+ω2 , and then compare it

to the compressed Pieri formula (4.18). The minimal coset representative of λ = 2ω1+ω2

is mλ = π∨s2s1s∨0 . Let µ = ω1. The six walks of type ~m−1

µ = (π∨)−1 beginning in m−1λ w−1

for w ∈ W0 are shown below; they begin on the sheet (π∨)2 and change sheets to end

on sheet π∨. The figure on the left shows where the walks begin, and the figure on the

right shows where the walks end. The left W0-cosets on each sheet are outlined in bold.

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Sheet (π∨)2

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•6

•5

•2

•3

•1

•4

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Sheet π∨

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........................................................................ ........................................................................↓6

↓5

↓2

↓3

↓1

↓4

Then Pω1P2ω1+ω2 =∑

h bhehP−w0wt(h), where the contributing coefficients of each walk is

given in the following table:

83

walk h −w0wt(h) bh eh d(h)

•1 2ω1 t3/21− q2

1− q2t

1− q3t

1− q3t21− q1− qt

t−1/2 1− q2t3

1− q2t2s2

•2 2ω1 t2/21− q3t

1− q3t21− q1− qt

t−2/2 1− q2t2

1− q2t

1− q2t3

1− q2t21

•3 3ω1 + ω2 t1/21− q1− qt

1 s2s1s2

•4 3ω1 + ω2 1 t−1/2 1− qt2

1− qts2s1

•5 ω1 + 2ω2 t1/21− q2

1− q2tt−2/2 1− q3t3

1− q3t21− qt2

1− qts1

•6 ω1 + 2ω2 t2/21− q2

1− q2t

1− q3t

1− q3t2t−1/2 1− qt2

1− qts1s2

Imposing the condition d(h) ∈ W ω1 = {1, s1, s2s1} on the final direction of each walk,

along with modified coefficients:

walk h −w0wt(h) bh eh d(h)

•2 2ω1 t2/21− q3t

1− q3t21− q1− qt

t−1/2 1− q2t3

1− q2t21

•4 3ω1 + ω2 1 1 s2s1

•5 ω1 + 2ω2 t1/21− q2

1− q2tt−1/2 1− qt2

1− qts1

84

formula (4.18) gives the compressed Pieri formula

Pω1P2ω1+ω2 = (t−1/2 + t1/2)

(P3ω1+ω2 +

1− q2

1− q2t

1− qt2

1− qtPω1+2ω2

+1− q3t

1− q3t21− q1− qt

1− q2t3

1− q2t2t1/2P2ω1

).

�

5.3 Type C2 examples

This example corresponds to the affine Type C2 in [30, (1.3.4)]. Let {ε1, ε2} be an

orthonormal basis for R2, and let {ε∨1 , ε∨2 } be its dual basis, where 〈εi, ε∨j 〉 = δij. The

simple roots and simple coroots of the complex simple Lie algebra sp4C are

α1 = ε1 − ε2, α2 = 2ε2, ϕ = ε1 + ε2, θ = 2ε1,

α∨1 = ε∨1 − ε∨2 , α∨2 = ε∨2 , ϕ∨ = ε∨1 + ε∨2 , θ∨ = ε∨1 ,

where ϕ∨ is the highest coroot, ϕ is the highest short root, θ is the highest root, and θ∨

is the highest short coroot. The root and coroot lattices are Q = Z(ε1 − ε2) + 2Zε2 and

Q∨ = Zε∨1 + Zε∨2 . The fundamental weights and fundamental coweights are

ω1 = ε1, ω2 = ε1 + ε2, ω∨1 = ε∨1 , ω∨2 = 12(ε∨1 + ε∨2 ) ,

and the weight and coweight lattices are h∗Z = Zε1 ⊕ Zε2 and hZ = Zε∨1 ⊕ 12Z(ε∨1 + ε∨2 ).

Note that only ω1 is a minuscule weight. The group Π∨ ∼= h∗Z/Q is the cyclic group of

order 2, and 〈h∗Z, hZ〉 ⊆ 12Z.

The Weyl group of this root system is the dihedral group of order eight

W0 = D4 =⟨s1, s2, | s1s2s1s2 = s2s1s2s1, s

2i = 1 for i = 1, 2

⟩,

85

and there are four reflections in the group D4, namely sα1 = s1, sα2 = s2, sϕ = s2s1s2,

and sθ = s1s2s1.

The simple affine root and coroot are α0 = −2ε1 + δ and α∨0 = −(ε∨1 + ε∨2 ) + d. Let

s∨0 = xε1+ε2s2s1s2, π∨ = xε1s1s2s1, s0 = yε∨1 s1s2s1, π = yω

∨2 s2s1s2.

Then the extended affine Weyl group W∨ is generated by D4 and the element π∨, subject

to the relations

(π∨)2 = 1, π∨s∨0 = s1π∨, π∨s2 = s2π

∨, s∨0 s1 = s1s∨0 , s∨0 s2s

∨0 s2 = s2s

∨0 s2s

∨0 ,

(5.14)

or alternatively,

W∨ = {xk1ε1+k2ε2w | k1, k2 ∈ Z, w ∈ D4}. (5.15)

The dual version of W∨ is the extended Weyl group W , which is generated by D4 and

π, subject the relations

π2 = 1, πs0 = s2π, πs1 = s1π, s0s1s0s1 = s1s0s1s0, s0s2 = s2s0, (5.16)

or alternatively,

W = {yk1ω∨1 +k2ω∨2 w | k1, k2 ∈ Z, w ∈ D4}. (5.17)

Figure 5.16 is the alcove picture for the extended affine Weyl group W∨, and Fig-

ure 5.17 is the (dual) alcove picture for the extended affine Weyl group W . Each shows

the correspondence between the alcoves and the elements of the respective group. The

periodic orientation is indicated by + and − on either side of the hyperplanes. The right

W0-cosets are indicated by bold lines.

86

Figure 5.16. Alcove picture for W∨.

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Hα∨2

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Hθ∨ .................................................................................................................................................................

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Hα∨1

+−

+−+

−

+−

Sheet π∨ •π∨ xω1x−ω1

xε2

x−ε2

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Hθ∨ ..................................................................................................................................................................................................................................................................................................................................

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Hϕ∨

Hα∨0

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.................................................................................................................................................................

Hα∨1

+−

+−+

−

+−

Sheet 1 •1

s1

s2

s1s2

s2s1

s1s2s1

s2s1s2

w0

s∨0

xα1

xα2

x−α1

x−θ

xϕ

x−ϕ

x−α2

87

Figure 5.17. Alcove picture for W .

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Hα2

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.................................................................................................................................................................

Hα1

+−

+−+

−

+−

Sheet π • π

yω∨2

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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Hθ Hα0.................................................................................................................................................................

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Hϕ.................................................................................................................................................................

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..................................................................................................................................................................................................................................................................................................................................

Hα1

+−

+−+

−

+−

Sheet 1 • 1

s1

s2w0sϕ

sθ s0 yε∨1

yα∨1

yε∨2 yϕ

∨

88

The double affine braid group B is generated by the groups

Π = 〈π | π2 = 1〉,

B = 〈T0, T1, T2 | T0T1 = T1T0, T0T2T0T2 = T2T0T2T0, T1T2T1T2 = T2T1T2T1〉,

{qkXk1ε1+k2ε2 | k ∈ 12Z, k1, k2 ∈ Z} ∼= h∗Z ⊕ 1

2Z,

subject to the relations

πT0π−1 = T1, πT2π

−1 = T2, πXε1π−1 = q1/2X−ε2 , πXε1+ε2π−1 = qX−ε1−ε2 ,

T1Xε1T1 = Xε2 , T1X

ε1+ε2 = Xε1+ε2T1, T2Xε1 = Xε1T2, T2X

ε1+ε2T2 = Xε1−ε2 ,

and the double affine Hecke algebra H over the field K = Q(q1/2, t1/21 , t

1/22 ) is the algebra

generated by the group algebra KB subject to the relations

T 2i = (t

1/2i − t

−1/2i )Ti + 1, for i = 0, 1, 2, where t0 = t2.

Define T∨1 = T1, T∨2 = T2,

(T∨0 )−1 = Xε1+ε2T2T1T2, π∨ = Xε1T1T2T1,

T0 = Y ε∨1 T−11 T−1

2 T−11 , π = Y

12

(ε1+ε2)T1T2T1.

The intertwiners are

π∨ = Xε1T1T2T1,

τ∨i = T∨i +t−1/2i − t1/2i

1− Y −α∨i= (T∨i )−1 +

(t−1/2i − t1/2i )Y −α

∨i

1− Y −α∨i, for i = 0, 1, 2,

where (π∨)2 = 1, π∨τ∨0 = τ∨1 π∨, π∨τ∨2 = τ∨2 π

∨, τ∨0 τ∨1 = τ∨1 τ

∨0 , τ∨0 τ

∨2 τ∨0 τ∨2 = τ∨2 τ

∨0 τ∨2 τ∨0 ,

and τ∨1 τ∨2 τ∨1 τ∨2 = τ∨2 τ

∨1 τ∨2 τ∨1 .

89

Chapter 6

Further work

In the Type A case, Lenart [22] compressed the alcove walk formula 3.14 to obtain a

tableau formula for symmetric Macdonald polynomials similar to the Haglund-Haiman-

Loehr formula [14], but with fewer terms. Several questions regarding “compression”

arise:

1. Can the Type A nonsymmetric alcove walk formula (Corollary 3.5) be compressed

to obtain a tableau formula for nonsymmetric Macdonald polynomials similar to

the Haglund-Haiman-Loehr formula [15]?

2. What kind of compression can be achieved for other root systems?

3. Is there a tableau version of the Littlewood-Richardson formula (Theorem 4.5)

that generalizes the Pieri formulas given in [27, (6.24)]?

4. Is there a tableau version of the Littlewood-Richardson formula (Theorem 4.1)

that generalizes the formulas for Demazure characters given in [16, Theorem 6.1]?

Gaussent and Littelmann [12] recently introduced the one-skeleton gallery model

to obtain a “geometric compression” of Schwer’s formula [37, Theorem 1.1] for Hall-

Littlewood polynomials, which is phrased in terms of positively folded galleries. So it is

natural to ask:

90

5. Is there a formulation of the Gaussent-Littelmann combinatorics in terms of double

affine Hecke algebras, and what kind of compression can it give to formulas for

Macdonald polynomials?

Another variation on the compression phenomenon is the following observation. Since

ε0H1 = ∆K[X]W01, then Proposition 2.13 implies that Aµ/∆1 is a W0-invariant poly-

nomial. In fact, Aρ1 = t|R+|/2∆1, and Equation [28, 7.3] can be rephrased as

Aλ+ρ(q, t) = Aρ(q, t)Pλ(q, qt).

This should be thought of as an analogue of the Weyl character formula.

6. What combinatorics can the alcove walk model reveal about the Weyl character

formula?

To conclude, Example 5.7 demonstrated that we have introduced enough tools to

be able to compute formulas of the form EµEλ =∑

ν cν(q, t)Eν . However, such a

formula is not “positive” in the sense that the numbers cν appearing in Eµ(0, 0)Eλ(0, 0) =∑ν cνEν(0, 0) are not positive (see [4] for example). The combinatorics arising from such

a Littlewood-Richardson rule for nonsymmetric Macdonald polynomials requires further

investigation.

91

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