a diagrammatic approach to categorifying quantum groups i

8/6/2019 A Diagrammatic Approach to Categorifying Quantum Groups I

1/53

arXiv:0803

.4121v2[math.QA

]5May2008 A diagrammatic approach to categorification

of quantum groups I

Mikhail Khovanov and Aaron D. Lauda

March 31, 2008

Abstract

To each graph without loops and multiple edges we assign a familyof rings. Categories of projective modules over these rings categorifyUq (g), where g is the Kac-Moody Lie algebra associated with the

graph.

Contents

1 Introduction 2

2 Rings R() and their properties 52.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 A faithful representation and a basis ofR() . . . . . . . . . . 152.4 Properties ofR() . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 Induction and restriction . . . . . . . . . . . . . . . . . . . . . 31

3 Quantum groups and the Grothendieck ring of R 343.1 Homomorphism of twisted bialgebras . . . . . . . . . . . . . 34

3.2 Surjectivity of . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Tight monomials and indecomposable projectives . . . . . . . 453.4 A conjecture on categorification of irreducible representations 48

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INTRODUCTION 2

1 Introduction

The goal of this paper is to categorify U = Uq (g), for an arbitrary simply-laced Kac-Moody algebra g. Here U stands for the quantum deformationof the universal enveloping algebra of the lower-triangular subalgebra of g.

Following the discovery of quantum groups Uq(g) by Drinfeld [12] andJimbo [18], Ringel [40] found a Hall algebra interpretation of the negativehalfU of the quantum group in the simply-laced Dynkin case. Lusztig [31],[32], [33] gave a geometric interpretation of U and produced a canonicalbasis there via a sophisticated approach which required the full strength ofthe theory of l-adic perverse sheaves. Kashiwara [19] defined a crystal basisofU at 0, a graph equipped with extra data, and in [20] constructed the so-

called global crystal basis ofU. Grojnowski and Lusztig [16] proved that theglobal crystal basis and the canonical basis are the same. The canonical basisB of U gives rise to bases in all irreducible integrable U-representations.Lusztig [34] also produced an idempotented version U of U and defined abasis there.

The work of Ariki [1] can be viewed as a categorification of the restricted

dual of U(g) for g = slN and g = slN and a categorification of all irre-ducible integrable representations of these Lie algebras (see also [27], [2], [3],[37]). An integral version of the restricted dual of U(g) becomes the sumof Grothendieck groups of suitable blocks of affine Hecke algebra representa-tions. An earlier work of Zelevinsky [47] can be understood in this contextas a parametrization of basis elements of U(g) via certain irreducible rep-resentations of affine Hecke algebras. Irreducible integrable representationsof U(g) become Grothendieck groups of Ariki-Koike cyclotomic Hecke alge-bras, which are certain finite-dimensional quotient algebras of affine Heckealgebras.

Grojnowski [14] found a purely algebraic way to understand these cate-gorifications via a generalization of Kleshchevs methods for studying modu-lar representations of the symmetric group [22], [23], [24]. This approach wasfurther developed by Grojnowski-Vazirani [17], Vazirani [45], [46], Brundan-Kleshchev [8] and others. It is explained in Kleshchev [25] in the context of

degenerate affine Hecke algebras.In this paper we introduce graded algebras categorifying Uq (g), for an

arbitrary simply-laced g. We start with an unoriented graph without loopsand multiple edges. Let I be the set of vertices of . The bilinear Cartan

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INTRODUCTION 3

form on [I] is given on the basis elements i, j I by

i j =

2 if i = j,

1 if i and j are joined by an edge,

0 otherwise.

The algebra U over (q), the negative (or positive) half of the quantumuniversal enveloping algebra, has generators i, i I, and defining relations

ij = ji if i j = 0,

(q + q1)iji = 2i j + j

2i if i j = 1.

The algebra U

contains a subring Af, which is the

[q, q1

]-lattice generatedby all products of quantum divided powers

(a)i . The canonical basis B is a

basis ofAf viewed as a free [q, q1]-module.

In Section 2 of this paper to each graph as above we assign a familyof graded rings R(), over [I]. The rings are defined geometrically,via braid-like plane diagrams which consist of interacting strings labelled byvertices of the graph. We prove basic results about these rings, then switchfrom the ground ring to a field k to simplify the study of R()-modules.We show that the representation theory of R() categorifies the integralform Af of U

. We consider the category R()pmod of finitely-generatedgraded left projective R()-modules and its Grothendieck group K0(R()).Let R = R() and define

K0(R) =

[I]

K0(R()).

Induction and restriction functors coming from the inclusions R()R() R(+) give rise to the multiplication and comultiplication homomorphisms

K0(R) K0(R) K0(R), K0(R) K0(R) K0(R)

that satisfy the same properties as those for Af. We define a homomorphism

of

[q, q1

]-algebras : AfK0(R) that also respects comultiplication andtakes a divided powers product element =

(a1)i1

. . . (ar )ir

to the image of acertain projective module P in the Grothendieck group.

The quantum Gabber-Kac theorem implies that is injective. By mir-roring for the case of rings R() the methods of Kleshchev, Grojnowski,

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INTRODUCTION 4

and Vazirani, who studied socles of induction and restriction applied to ir-

reducible representations, we show that homomorphism is surjective forany graph and any field k. The main result of the paper is the followingtheorem.

Theorem 1.1. : AfK0(R) is an isomorphism of [I]-graded twistedbialgebras.

The term twisted bialgebras is used above, since the comultiplicationin Af and K0(R) becomes an algebra homomorphism only after the multi-plication in the tensor squares Af

2 and K0(R)2 is twisted by powers of

q.We conjecture that, when is a tree and k = , isomorphism takes

canonical basis elements to the images of indecomposable projective modulesin K0(R). When the graph is a single vertex, this conjecture is almost trivial.We verify the conjecture in a simple case of the graph with two verticesand one edge, with all canonical basis elements being monomials.

The rings R() should be linked to Lusztigs geometric realization ofAf:

Conjecture 1.2. For k = and graph a tree, the algebra R() is Moritaequivalent to the algebra of equivariant ext groups ExtG (L, L), where G =

iI GL(i) and L is the sum of simple perverse sheaves Lb, over all b B,in Lusztigs geometric realization ofAf.

This conjecture should follow from an isomorphism between R() and asuitable convolution algebra. When contains cycles, its possible to modifyR() by introducing monodromies around the cycles, and the conjectureis likely to hold for a modified version of R().

Our results hint at the relation between representation theory of affineHecke algebras for GL(n) when q is not a root of unity and representations ofR() when the graph is a chain. We conjecture that completions of affineHecke algebras along suitable maximal central ideals are Morita equivalent(or even isomorphic) to completions of R() along the grading. The aboveconjectures, if true, would link Lusztigs geometrization of U with Arikis

categorification of U

and its restricted dual for q = 1 and = An.We arrived at the definition of rings R() from computations involvinghomomorphisms of bimodules over cohomology rings of partial flag varieties.The bimodules themselves are the cohomology groups of partial and iteratedflag varieties that give correspondences for the action of generators ei and

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RINGS R() AND THEIR PROPERTIES 5

fi of quantum slN in the Beilinson-Lusztig-MacPherson geometric model [4]

of quantum slN. This model was given a categorical interpretation by Gro- jnowski and Lusztig [15] and later reinterpreted, for N = 2, via transla-tion and Zuckerman functors in [6], with various generalizations constructedin [13], [44], [48], and in a very recent striking work [49].

In Section 3.4 we define certain quotient algebras of R() and conjecturethat their categories of modules categorify irreducible integrable represen-tations of Uq(g). A straightforward generalization of our constructions andresults from algebras R() and their quotient algebras in the simply-lacedcase to that of an arbitrary symmetrizable Kac-Moody algebra g will bepresented in a follow-up paper.

In the paper [9] (see also [41]), Chuang and Rouquier defined sl(2)-

categorifications, substantiated them with many diverse examples, and ap-plied to the modular representation theory. Partially inspired by [9] and [10],the second author suggested and investigated a categorification of Lusztigsidempotent completion U of quantum sl(2) in [28], [29]. A definition of acategorification ofUq(g) for any simply-laced g can be obtained by combiningthe diagrammatic relations ofR() with those of U-categorification in [28].

R. Rouquier [42], in his recent talk, defined sl(N)- and affine sl(N)-categorifications and outlined a conjectural program that aims to vastly gen-eralize his prior work with J. Chuang [9] on sl(2)-categorifications. We expectRouquiers and our approaches to be closely related. R. Rouquier informed

us that a signed version of rings R() appears in his categorification [43] ofU(g) for a simply-laced g.

Acknowledgments: This paper was written while the first author wasat the Institute for Advanced Study, and he would like to thank the Institutefor its hospitality and the NSF for fully supporting him during that timethrough the grant DMS-0635607. Additional partial support came from theNSF grant DMS-0706924.

2 Rings R() and their properties

2.1 Definitions

We fix a graph , not necessarily finite, with set of vertices I and unorientededges E. We require that has no loops and multiple edges. By [I] wedenote the commutative semigroup freely generated by vertices of and for

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DEFINITIONS 6

[I] write

= iI

i i , i , = {0, 1, 2, . . . }. (2.1)

Let || =

i , and Supp() = {i | i = 0}. We define a bilinear formon [I] by i i = 2, i j = 1 if i and j are connected by an edge, andi j = 0 otherwise. In the basis {i}iI of vertices the bilinear form is givenby the Cartan matrix of .

To we associate a diagrammatic calculus of planar diagrams. We con-sider collections of arcs on the plane connecting m points on one horizontalline with m points on another horizontal line. The position of m points

on the line is fixed once and for all (for instance, we could take points{1, 2, . . . , m} ). Arcs have no critical points when projected to the y-axisof the plane (a condition reminiscent of braids). Each arc is labelled by avertex of . Arcs can intersect, but no triple intersections are allowed. Anarc can carry dots. An example of such a diagram is shown below,

i j i k

(2.2)

where i,j,k are vertices of and the label of an arc is written at the bottomend of the arc. We allow isotopies that do not change the combinatorial typeof the diagram and do not create critical points for the projection onto thez-axis.

i j i kk

i j i kk

We proceed by allowing finite linear combinations of these diagrams with

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DEFINITIONS 7

integral coefficients, modulo the following local relations

i j

=

0 if i = j,

i j

if i j = 0,

i

j

+

j

i

if i j = 1.

(2.3)

i j

= i j

i j

= i j

for i = j (2.4)

i i

i i

=

i i

(2.5)

i i

i i

=

i i

(2.6)

i j k

=

i j k

unless i = k and i j = 1 (2.7)

i j i

i j i

=

i j i

if i j = 1 (2.8)

Fix [I]. Let Seq() be the set of all sequences of vertices i = i1 . . . imwhere ik I for each k and vertex i appears i times in the sequence. The

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DEFINITIONS 8

length m of the sequence is equal to || and the cardinality of Seq() is equal

to i, j , . . ., taken over all i I. For instance,Seq(2i + j) = {iij, iji, jii}.

Each diagram D as described above determines two sequences bot(D) andtop(D) in Seq() for some . The sequence bot(D) is given by reading thelabels of arcs ofD at the bottom position from left to right. We define top(D)likewise. For instance, for the diagram in (2.2), bot(D) = ijik and top(D) =

jiki. We often abbreviate sequences with many equal consecutive terms, andwrite in11 . . . i

nrr for i1 . . . i1i2 . . . i2 . . . ir . . . ir, where n1 + + nr = m.

Define the ring R() as follows:

R() =

i,jSeq()

jR()i (2.9)

as an abelian group, where jR()i is the abelian group of all linear combi-nations of diagrams with bot(D) = i and top(D) = j modulo the relations(2.3)(2.8). The product in R() is given by concatenation (see the leftdiagram in (2.11) below)

kR()j jR()i kR()i, (2.10)

and xy = 0 for x lR()k and y jR()i if k = j.

k

j

i i1 i2

. . .

im

(2.11)

By construction, R() is an associative ring. For each i Seq() thediagram 1i iR()i shown on the right of (2.11) is an idempotent, 12i = 1i,x1i = x for all x jR()i and 1ix = x for all x iR()j, for all j.

Furthermore, 1 = iSeq() 1i is the unit element of R(). We turn R()into a graded ring by declaring degrees of the generators to bedeg

i

= 2, deg

i j

= i j. (2.12)8


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DEFINITIONS 9

Let

Pi = jSeq()

jR()i, jP = iSeq()

jR()i. (2.13)

Pi is a left graded projective R()-module and jP is a right graded projectiveR()-module.

Flipping a diagram about a horizontal axis induces a gradingpreservingantiinvolution ofR() which takes jR()i to iR()j and 1i to 1i. Flippinga diagram about a vertical axis and simultaneously taking

i i

to

i i

(in other words, multiplying the diagram by (1)s where s is the numberof times equally labelled strands intersect) is an involution of R() whichcommutes with .

Sometimes it is convenient to convert from graphical to algebraic notation.For a sequence i = i1i2 . . . im Seq() and 1 k m we denote

xk,i :=

i1

. . .

ik

. . .

im

(2.14)

with the dot positioned on the k-th strand counting from the left, and

k,i :=

i1

. . .

ik ik+1

. . .

im

(2.15)

The symmetric group Sm acts on Seq(), m = || by permutations. Trans-position sk = (k, k + 1) switches entries ik, ik+1 of i. Thus, k,i sk(i)R()i.The relations (2.3) become

k,sk(i)k,i = 0 if ik = ik+1.1i if ik ik+1 = 0

xk,i + xk+1,i if ik ik+1 = 1

(2.16)

Other defining relations for R() can be similarly rewritten.

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EXAMPLES 10

2.2 Examples

1) = 0. We have R(0) = , with the unit element given by the emptydiagram.

2) = i for some vertex i. Then a diagram is a line with some number ofdots on it. Hence, R(i) = [x1,i], where in our notation x1,i denotes a linelabelled i with one dot on it.

i

3) = mi for some vertex i. The only sequence in Seq(mi) is im = i i . . . i.

Every strand in the diagram is labelled by i, and the local relations are

i i

= 0

i i i

=

i i i

(2.17)

i i

i i

=

i i

(2.18)

i i

i i

=

i i

(2.19)

R(mi) is isomorphic to the nilHecke ring NHm, which is the unital ringof endomorphisms of the abelian group [x1, . . . , xm] generated by theendomorphisms of multiplication by x1, . . . , xm and divided difference op-erators

a(f(x)) =f(x) saf(x)

xa xa+1, 1 a m 1,

where sa transposes xa and xa+1 in the polynomial f(x). The defining

relations arexaxb = xbxa,axb = xba if |a b| > 1, ab = ba if |a b| > 1,2a = 0, aa+1a = a+1aa+1,xaa axa+1 = 1, axa xa+1a = 1.

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EXAMPLES 11

In the above equations xa stands for the operator of multiplication by xa.

The defining relations are exactly our graphical equations on identically-colored strands, with the relations in the first two rows corresponding toisotopies of diagrams.

The nilHecke ring is related to the theory of Schubert varieties, see [26],[7]. The nilCoxeter ring is the subring ofNHm generated by 1, . . . , m1,see [38, Chapter 2], [21]. Divided differences go back to Newton; in thecontext of representation theory they appeared in [5], [11].

The center of NHm is the ring of symmetric polynomials in x1, . . . , xm,and NHm is isomorphic to the ring ofm! m! matrices with coefficientsin Z(NHm), see [36]. Here we consider NHm as a graded ring, with

deg(a) = 2 and deg(xa) = 2. The graded nilHecke ring plays a fun-damental role in the categorification of Lusztigs quantum sl2 defined bythe second author [28].

For each permutation w Sm let w = a1 . . . ar , where sa1 . . . sar isa minimal presentation of w, so that r = l(w). This element does notdepend on the choice of presentation.

Define em = xm11 x

m22 . . . xm1w0 , where w0 is the longest permutation.

This element is an idempotent of degree 0. We will also use the idempotent(em) given by reflecting the diagram of em about the horizontal axis,

(em) = w0x

m1

1 x

m2

2 . . . xm1.

NHm(em) is a left NHm-module isomorphic to the polynomial repre-sentation of NHm. The polynomial representation is the unique, up toisomorphism and grading shifts, graded indecomposable projective NHm-module. The module NHm(em) is nontrivial in even non-negative de-grees only.

The regular representation of NHm decomposes as the sum of m! copiesof the polynomial one. Taking the grading into account and denoting byPm the module NHm(em) with the grading shifted down by

m(m1)2

, weget a direct sum decomposition of graded modules

NHm = P[m]!

m .

Here [m]! = [m][m 1] . . . [1] is the quantum factorial, [m] = qmqm

qq1,

and Mf or Mf, for a graded module M and a Laurent polynomial f =

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EXAMPLES 12

faq

a [q, q1], denotes the direct sum over a , of fa copies of

M{a}.We denote by Pi,m the corresponding indecomposable graded projectivemodule over R(mi) and by ei,m the idempotent corresponding to em underthe isomorphism R(mi) = NHm. As a graded abelian group, Pi,m is

nontrivial in degrees m(m1)2

+ 2 .

Letting mP be the right graded projective module emNHm{m(m1)

2}, we

have a decomposition of graded right NHm-modules

NHm = mP[m]!.

We denote by i,mP the corresponding indecomposable graded projectiveright R(mi)-module. Idempotents ei,m and (ei,m) have the followingdiagrammatic presentation for m = 3

ei,3 =

i i i

(ei,3) =

i i i

(2.20)

The quotient of [x1, . . . , xm] by the ideal of symmetric polynomials isa representation Lm of NHm which becomes irreducible upon tensor-

ing with any field k. Over k, any graded irreducible representation ofNHm is isomorphic to Lm, up to a grading shift. We denote the corre-sponding irreducible representation of R(mi) by L(im). Its nonzero in

degrees 0, 2, . . . , m(m1)2

. The representation L(im) is isomorphic to therepresentation induced from the one-dimensional graded module L overk[x1, . . . , xm] (on which x1, . . . , xm necessarily act trivially).

Lemma 2.1. The common 0-eigenspace of operators x1, . . . , xm on L(im) =

Ind(L) is exactly 1 L. All Jordan blocks of xm on L(im) are of size m.

Proof. Due to the uniqueness of the irreducible module L(im), it is isomor-

phic to the module induced from the trivial representation of the subringof NHm generated by the divided differences 1, . . . , m and symmetricpolynomials in x1, . . . , xm. This induced representation is isomorphic,as a k[x1, . . . , xm]-module, to the quotient of k[x1, . . . , xm] by the idealgenerated by symmetric polynomials without the constant term, and to

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EXAMPLES 13

the cohomology ring of the full flag variety. The lemma follows from the

standard facts about this quotient ring.

4) = i+j and ij = 0. Seq(i+j) = {ij,ji}. The ring R() is isomorphic tothe ring of 2 2 matrices with coefficients in [x1, x2]. The isomorphismis given on generators by

i j

1 00 0

j i

0 00 1

ij

0 01 0

j

i

0 10 0

i

j

x1 0

0 0

i j

x2 0

0 0

5) = i1 + + im and ik i = 0 for all k = . Then R() is isomorphicto the ring ofm! m! matrices with coefficients in [x1, . . . , xm]. To seethe isomorphism, enumerate the rows and columns by elements of Seq()and send the element j1i to the elementary (j, i)-matrix.

6) = + such that i j = 0 for any i Supp() and j Supp().

In this case R() is isomorphic to the matrix algebra of size ||||, ||with coefficients in R()

R(). Indeed, except for crossings, thereare no interactions between strands from and strands from . A pairi Seq(), j Seq() defines the sequence i j Seq( + ) and

i jRi j = R() R(), (2.21)

since we can pull apart the i and j strands in any diagram D with i j =top(D) = bot(D), using that ik j = 0, for all k and .

7) = i + j and i j = 1. We can identify R(i + j) with the ring of 2 2

matrices with coefficients in [x1, x2] such that the bottom left coefficient

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EXAMPLES 14

is divisible by x1 + x2:

i j

1 0

0 0

j i

0 0

0 1

i j

0 0

x1 + x2 0

j i

0 10 0

i

j

x1 00 0

i j

x2 00 0

Remark 2.2. If i j = 1 then the elements

i j i

and

i j i

(2.22)

are mutually orthogonal idempotents in R(2i + j). For instance,

i j i

2

=

i j i

(2.3)

i j i

+

i j i

(2.3)

i j i

(2.5)

i j i

+

i j i

(2.3)

i j i

(2.23)

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A FAITHFUL REPRESENTATION AND A BASIS OF R() 15

Equation (2.8) can be viewed as a decomposition of the idempotent 1iji

into the sum of two orthogonal idempotents. Equation (2.7) can bethought of as allowing triple intersections for certain ijk.

2.3 A faithful representation and a basis of R()

Action ofR() on the sum of polynomial spaces. Choose an orientationof each edge of . For each [I] we define an action of R() on the freeabelian group

Po =

iSeq()

Poi, Poi = [x1(i), x2(i), . . . , xm(i)], m = ||.

Its useful to think of the variable xk(i) as labelled by the vertex of in thek-th position in the sequence i. The symmetric group Sm acts on Po bytaking xa(i) to xw(a)(w(i)), w Sm. The transposition sk maps xa(i) toxa(sk(i)) if a = k, k + 1, xk(i) to xk+1(ski), and xk+1(i) to xk(ski).

To define the action of R(), we first require that an element x jR()iacts by 0 on Pok ifk = i and takes Poi to Poj. We describe the action ofthe generators. The dot in the k-th position xk,i (see (2.14)) acts by sendingf Poi to xk(i)f Poi. The idempotent 1i act by the identity on Poi.The crossing k,i (see (2.15)) acts on f Poi by

f skf if ik ik+1 = 0,

f f skf

xk(i) xk+1(i)if ik = ik+1,

f skf if ik ik+1,

f (xk(ski) + xk+1(ski))(skf) if ik ik+1.

Notation ik ik+1 means that ik ik+1 = 1 and this edge of is orientedfrom ik+1 to ik. Note that when ik = ik+1 the crossing k,i acts by the divideddifference operator. When all strands have the same label i, the actionreduces to the action of the nilHecke algebra on its polynomial representation.

Proposition 2.3. These rules define a left action of R() on Po.

Proof. We check the defining relations for R(). The relation (2.3) withij = 0 and ij = 1 and relation (2.4) are trivial to verify. The relation (2.3)with i = j and relations (2.5), (2.6), and (2.7) for i = j = k are just

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the nilHecke relations. The relation (2.7) with i,j,k all distinct, or with

i = j = k, i j = 0, or with i = j = k, j k = 0 is easy to check. Thesame relation with i = j, i k = 1 or j = k, i j = 1 follows from thefact that the divided difference operator annihilates symmetric polynomials.This leaves us with the last relation (2.8), reproduced below

i j i

i j i

=

i j i

if i j = 1.

Its enough to check it on 3-stranded diagrams, with = 2i +j. To simplifyformulas, we write x,y, and z instead ofxk(i), xk+1(i), and xk+2(i) for eachi = {iji, iij, jii}. Assume that the ij edge is i j. The left hand side ofthe relation is

1,jii2,jii1,iji 2,iij 1,iij2,iji,

taking Poiji to Poiji . We compute the action of this element on eachmonomial xuyvzw, u,v,w :

1,jii2,jii1,iji(xuyvzw) = 1,jii2,jii(x

vyuzw) = 1,jii

xvyuzw xvywzu

y z

=

xuyvzw xwyvzu

x z (x + y), (2.24)2,iij 1,iij2,iji(x

uyvzw) = 2,iij1,iij ((y + z)xuywzv)

= 2,iij

xuyw+1 xw+1yu

x yzv +

xuyw xwyu

x yzv+1

= 2,iij

xuyw+1zv xw+1yuzv + xuywzv+1 xwyuzv+1

x y

=

xuyvzw+1 xw+1yvzu + xuyv+1zw xwyv+1zu

x z. (2.25)

One can easily verify that the difference of (2.24) and (2.25) is xuyvzw, prov-

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ing relation (2.8) in this case. When i j, we compute

1,jii2,jii1,iji(xuyvzw) = 1,jii2,jii((x + y)x

vyuzw)

= 1,jii

xv+1

yuzw ywzu

y z+ xv

yu+1zw ywzu+1

y z

= 1,jii

xv+1yuzw xv+1ywzu + xvyu+1zw xvywzu+1

y z

=

xuyv+1zw xwyv+1zu + xu+1yvzw xwyvzu+1

x z, (2.26)

2,iij 1,iij2,iji(xuyvzw) = 2,iij1,iij (x

uywzv) = 2,iij

xuywzv xwyuzv

x y

= xuyvzw xwyvzu

x z(y + z). (2.27)

Again, the difference of (2.26) and (2.27) is xuyvzw. Relation (2.8) andProposition 2.3 follow.

A spanning set. We look for a lower bound on the size of R(). Anelement of this ring is a linear combination of diagrams.

If a diagram D contains two strands that intersect more than once, re-lations (2.3)(2.8) allow us to write D as a linear combination of diagrams,each with fewer intersections than D. Iterating, we can write any element

of R() as a linear combination of diagrams with at most one intersectionbetween any two strands. Furthermore, we can slide all dots in a diagramD all the way to the bottom of the diagram at the cost of adding a linearcombination of diagrams with fewer crossings than D. These two operationstogether tell us that R() is spanned by diagrams having all dots at thebottom and with each pair of strands intersecting at most once:

Such D is determined by i = bot(D), a minimal presentation w =sk1 . . . skr , r = l(w) of a permutation w Sm and the number of dots at

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each bottom endpoint of D. The difference of two diagrams given by the

same data except for different minimal presentations of w can be writtenas a linear combination of diagrams with fewer crossings than each of theoriginal two diagrams.

For each w Sm fix its minimal presentation w. For i,j Seq() letjSi be the subset of Sm consisting of permutations w that take i to j viathe standard action of permutations on sequences, defined earlier. For eachw jSi we convert its minimal presentation w into an element of jR()idenoted wi. Denote the subset {wi}wjSi of jR()i by jSi.Example 2.4. iji Siji = {id, (13)}, and

idiji =i j i

, (13)iji =i j i

or

i j i

,

depending on whether we choose s2s1s2 or s1s2s1 as a minimal presentationof permutation (13).

In general, jSi depends on our choices of minimal presentations for per-mutations. For instance, in the above example,

(13)iji iji

Siji will de-

pend nontrivially on whether the presentation s1s2s1 or s2s1s2 was chosen if

i j = 1.Let jBi be the set {y x

u11,i . . . x

umm,i} over all y j

Si and ui . Here thediagrams in jSi are multiplied by all possible monomials at the bottom. Forexample, the sets ijBij and jiBij consist of elements

i

j

u1 u2 and i j

u1 u2

respectively, where we write

u = u .Theorem 2.5. jR()i is a free graded abelian group with a homogeneousbasis jBi.

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Proof. We have already observed that the set jBi spans jR()i. This set

consists of homogeneous elements relative to our grading on R(). To provelinear independence of elements of jBi we check that they act on Po bylinearly independent operators.

Let j1i be the diagram with the fewest number of crossings with bot(j1i) =i and top(j1i) = j. For example,

jjiki1ijkij =

i j k i j

j j i k i

Note that identically coloured lines do not intersect in j1i, and that i1i isjust 1i.

The product i1j j1i =

(xa,i + xb,i) where the product is over all pairs1 a < b m such that the lines in j1i ending at a and b bottom endpointscounting from the left intersect and are coloured by i, j with i j = 1.

For instance, if i = ijj , j = jj i with i j = 1, then

j1i =

i j j

j j i

(2.28)

and the product

i1j j1i =

i j j

= (x1,i + x2,i)(x1,i + x3,i). (2.29)

Choose a complete order on the set of vertices of and orient so thatfor each edge i j we have i < j relative to the order. This order inducesa lexicographic order on Seq(). We prove linear independence of jBi byinduction on j Seq() with respect to this order.

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Base of induction: We write

j = j1 . . . j1j2 . . . j2 . . . jr . . . jr = j11 j

22 . . . j

rr Seq(), =

rk=1

kjk,

(2.30)where j1 < j2 < < jr and r is the cardinality of Supp(). Clearly j is thelowest element in Seq() with respect to lexicographic order. For this j eachw jSi can be written uniquely as w = w1w0 where w1 S1 Srand w0 is the unique minimal length element in jSi.

Each minimal length representative w0 determines the same w0i = j1i.Likewise, the element

w1j does not depend on the choice of a minimal length

representative w1, since in the nilHecke algebra the element associated to apermutation does not depend on the minimal presentation of this permuta-tion.

j1 j1 j1 j2 j2 jr jr jr

w1

w0

The set jBi consists of elements w1jw0ixu, over all u m, wherexu = xu11,ix

u22,i . . . x

umm,i.

It suffices to check that induced maps

w1jw0ixu : Poi Poj (2.31)are linearly independent, over all u m. Indeed,

w0ix

u takes xv Poito xw0(u+v) Poj, where w0 acts on v via the obvious permutation. This

is due to peculiarities of our action, since the element k,i takes f Poi toskf if ik > ik+1. Elements w1j act on the monomials by products of divideddifference operators. Its known that the standard action of the nilHeckering on polynomials is faithful [36], implying linear independence of all maps(2.31).

20


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Induction step: Assume we proved that jBi is independent, that jk 0. Thensoc(eiM) is irreducible and i(soc(eiM)) = i(M) 1. Socles of eiM arepairwise non-isomorphic for different i I.

Proof is the same as for Corollaries 5.1.7 and 5.1.8 in [25]. For an irreducible M R()mod define

eiM := soc(eiM), fiM := hd ind+i,i M L(i). (3.3)The module fiM is irreducible by Lemma 3.9, while eiM is irreducible or 0by Corollary 3.12, and

i(M) = max{n 0|eni M = 0}, i( fiM) = i(M) + 1.In the statements below, isomorphisms of simple modules are allowed to

be homogeneous (not necessarily degree-preserving).

Lemma 3.13. For an irreducible M R()mod we have

socin M = (

eni M) L(i

n), (3.4)

hd ind,ni(M L(in)) = f

ni M. (3.5)

Lemma 3.14. For an irreducible M R()mod the socle of eni M is iso-

morphic to eni M[n]!{ n(n1)2 }.The proofs are equivalent to those of Lemmas 5.2.1 and 5.2.2 in [25].

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SURJECTIVITY OF 42

Lemma 3.15. For irreducible modules M R()mod and N R( +

i)mod we have fiM = N if and only ifeiN = M.Proof follows that of Lemma 5.2.3 in [25].

Corollary 3.16. Let M, N R()mod be irreducible. Then fiM = fiNif and only ifM = N. Assuming i(M), i(N) > 0, eiM = eiN if and only ifM = N.

The character ch(M) of a finite-dimensional representation M R()modtakes values in [q, q1]Seq(), the free [q, q1]-module generated by Seq(),and descends to a homomorphism from the Grothendieck group G0(R()) to [q, q1]Seq().

Theorem 3.17. The character map

ch : G0(R()) [q, q1]Seq()

is injective.

Equivalently, the characters of irreducible modules (one from each equiva-lence class up to grading shifts) are linearly independent functions on Seq().The proof is identical to that of Theorem 5.3.1 in [25]. Note that in our casethe character of a finite-dimensional graded module is a function on sequenceswith values in [q, q1], while in the non-graded case of [25] its a function on

sequences taking values in . This discrepancy has no effect on the proof. Passing to the fraction field (q) of [q, q1] and dualizing the map ch,

which then becomes the composition

(q)Seq()f

(q) K0(R())

(q),

we conclude that

(q), restricted to weight , is a surjective map of (q)-vector spaces. We have already observed that and

(q) are injective. Bysumming over all weights, we obtain the following result.

Proposition 3.18.

(q) : fK0(R) (q) is an isomorphism.

Therefore, the number of isomorphism classes of (graded) simple R()-modules is the same for any field k.

Corollary 3.19. A (graded) irreducible R()-module is absolutely irreducible,for any , k and weight .

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SURJECTIVITY OF 43

Next, assume that is finite. Choose a total order on I, i(0) < i(1) k define i(r) = i(r

) where r

is the residueof r modulo k. Fix one representative Sb from each isomorphism class bof irreducible R()-modules, up to grading shifts. Recall that we denotedthis set of isomorphism classes by B. For all , to each b B

assign

the following sequence Yb = y0y1 . . . of nonnegative integers: y0 = i(0)(M),and let M1 = ey0i(0)M. Inductively, yr = i(r)(Mr), and Mr+1 = eyri(r)Mr.Note that y0 + y1 + = || and only finitely many terms in the sequenceare non-zero. Introduce a lexicographic order on sequences of non-negativeintegers: y0y1 > z0z1 . . . if, for some t, y0 = z0, y1 = z1, . . . yt1 = zt1and yt > zt. This order induces a total order on B

, by b > c iff Yb > Yc.

To each sequence Yb = y0y1 . . . we assign the projective R()-module PYrb

associated to the divided powers sequence Yrb = . . . i(2)(y2)i(1)(y1)i(0)(y0) (the

order of ys is reversed).

Proposition 3.20. HOM(P(Yb), Sc) = 0 ifb < c and HOM(P(Yb), Sb) = k.

This follows from the previous results and implies that the image [P] ofany (graded) projective R()-module in the Grothendieck group K0(R())can be written as a linear combination, with coefficients in [q, q1], of imagesof divided powers projectives [P], for divided power sequences of the formYrb . Therefore, : AfK0(R()) is surjective. Since, is also injective,it is an isomorphism. The case of an infinite follows by taking the direct

limit of its finite subgraphs. This concludes the proof of the Theorem 1.1stated in the introduction.

It would be interesting to find out if the theorem remains valid for ringsR() over rather than over a field k.

For each divided power i(a) we have the corresponding projective Pi(a). In-

duction with this projective is an exact functor, denoted F(a)i , from R()mod

to R(+ ai)mod. Summing over all , form the functor

F(a)i : RmodRmod.

This functor restricts to the subcategory Rpmod of the category of projec-tive modules. To any divided power sequence = i

(a1)1 . . . i

(ar )r associate the

functorF = F

(a1)i1

F (ar )ir

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SURJECTIVITY OF 44

on Rmod. To a finite sum

kuk(k) where uk [q, q

1] and (k) are

divided powers sequences associate the direct sum of shifted copies of F(k):k

Fuk(k) .

Theorem 3.21. For any relationk

uk(k) =

v()

in Af with positive coefficients uk, v [q, q1] there is an isomorphism of

projectives k

Puk(k) =

Pv()

inducing an isomorphism of functorsk

Fuk(k)

=

Fv().

This result follows immediately from the earlier ones. We conclude that any relation in Af lifts to an isomorphism of functors. It

is natural to view the category Rpmod, as well as the category of inductionfunctors on Rmod it gives rise to, as a categorification of Af, the integralform of the quantum universal enveloping algebra of the negative half of thesimply-laced Kac-Moody algebra associated to the graph .

The semilinear hom form (, ) on K0(R) defined by

([P], [Q]) := gdimHOM(P, Q)

is related to the tensor product bilinear form (, ) given by (2.45) via

(x, y) = (x, y).

Indeed, by surjectivity of , it suffices to check this relation for x = [Pi]and y = [Pj], in which case both sides are equal to the graded dimension of

iR()j.The involution ofR() defined in Section 2.1 induces a self-equivalence

of R()mod which takes projective Pi(a1)1 ...i

(ar)r

to Pi(ar)r ...i

(a1)1

. The induced

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TIGHT MONOMIALS AND INDECOMPOSABLE PROJECTIVES 45

map [] on the Grothendieck group K0(R) coincides, under the isomorphism

, the q-linear anti-involution ofAf that fixes each (a)

i .On the category R()fmod we have the contravariant duality functor

which takes a finite-dimensional module M to its vector space dual M

twisted by the antiinvolution . This duality functor leaves invariant thecharacter evaluated at q = 1:

ch(M)q=1 = ch(M)q=1.

Therefore, the contravariant duality preserves simples, up to overall shift:

Sb= Sb{r}.

Given i Seq(), we have ch(Sb, i)

[q2

, q2

] or ch(Sb, i) q

[q2

, q2

] forparity reasons (more generally, this is true for any indecomposable object ofR()mod and can be used to decompose R()mod into the direct sumof two subcategories). Then ch(Sb , i) [q

2, q2] if the same is true for

ch(Sb, i), and ch(Sb , i) q [q

2, q2] if ch(Sb, i) q [q2, q2]. Hence, the

shift r is an even number.From now on we redefine Sb by shifting its grading by

r2 . We have S

b

=Sb as graded modules. This normalization of Sb does not depend on thechoice of i. The character ofSb is bar-invariant:

ch(Sb, i) = ch(Sb, i)

for all i Seq(), where q = q1. Extending the bar-involution to [q, q1]Seq()by i = i, we have ch(Sb) = ch(Sb).

This canonical (balanced) choice of grading for Sb allows us to fix thegrading on indecomposable projective Pb so that the quotient map PbSbis grading-preserving. In this way we obtain a basis {[Pb]} in Af whichdepends only on the characteristic of k. Both the multiplication and thecomultiplication in this basis have coefficients in [q, q1]. An example belowshows this basis to be different from the Lusztig-Kashiwara basis when isan odd length cycle and k has any characteristic, and when is a cycle andk has characteristic 2.

3.3 Tight monomials and indecomposable projectives

Following Lusztig [35], we say that a monomial = (a1)1 . . .

(ak )k is tight if it

belongs to the canonical basis B ofAf. It follows from the properties of the

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canonical basis that a monomial is tight if and only if (, ) 1 q [q] (or

see [39, Proposition 3.1]).Proposition 3.22. If a monomial is tight, the projective module P isindecomposable.

Proof. Tightness of implies that HOM(P, P) is a +-graded k-vector spacewhich is one-dimensional in degree 0. Therefore, any degree 0 endomorphismof P is a multiple of the identity, and P is indecomposable. This argumentworks even over .

Example 3.23. When the graph consists of a single vertex i, the weightspace Afmi is a rank one free [q, q

1]-module generated by (m)i . The map

takes it to [Pi(m) ], the generator of the Grothendieck group K0(R(mi)).Projective module Pi(m) is indecomposable.

Example 3.24. Let = i j

. Tight monomials (a)i

(b)j

(c)i (a,b,c ,

b a + c) and (c)

j (b)i

(a)j (a,b,c , b a + c), with the identification

(a)i

(a+c)j

(c)i =

(c)j

(a+c)i

(a)j ,

constitute the canonical basis B ofAf, see [34, Example 14.5.4]. Therefore,images of indecomposable projectives Pi(a)j(b)i(c) , b a + c and Pj(c)i(b)j(a),

b > a + c, constitute a basis in the free

[q, q

1

]-module K0(R). Any inde-composable projective in Rmod is isomorphic to one of the above, up to agrading shift. Indecomposables Pi(a)j(a+c)i(c) and Pj(c)i(a+c)j(a) are isomorphic.

Example 3.25. Let =

///

/// be a cycle with n 3 vertices. Label the

vertices clockwise by 1, 2, . . . , n and let i = 12 . . . n. Then HOM(Pi i, Pi i)is +-graded and Hom(Pi i, Pi i) is 2-dimensional with the basis {i i1i i, },where

=

1 2 3 n 1 2 3 n

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A computation shows that deg() = 0 and

2 = 0 if n is odd,

2 if n is even;

implying that i i1i i is the only idempotent in Pi i if n is odd, or if n is evenand char(k) = 2. Under these assumptions, Pi i is indecomposable, but

[Pi i], [Pi i]

2 + q [q] = 1 + q [q],

and [Pi i] is not a canonical basis element. For an odd length cycle we foundan indecomposable projective Pi i whose image in the Grothendieck group isnot a canonical basis vector, while being invariant under the bar involution:[Pi i] = [Pi i].

When n is even and char(k) = 2, 0 = 2 is an idempotent in Hom(Pi i, Pi i),

and Pi i = Pi i0 Pi i(1 0) is isomorphic to the direct sum of two inde-composable projectives. Furthermore, = 10 where

1 =

1 1 2 2 3 3 n n

0 =

1 2 3 n 1 2 3 n

Module homomorphisms

1 : Pi i P1(2)2(2)...n(2),

0 : P1(2)2(2)...n(2) Pi i,

induced by these elements via right multiplication have degree 0, and 01 =2 Id, so that Pi i0 = P1(2)2(2)...n(2).

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A CONJECTURE ON CATEGORIFICATION OF IRREDUCIBLE

REPRESENTATIONS 48

3.4 A conjecture on categorification of irreducible rep-

resentationsChoose [I], =

i i, i I. Let R(; ) be the quotient ring of

R() by the ideal generated by all diagrams of the form

i1 i2

. . .

im

i1

where i1 . . . im Seq() and the leftmost string has i1 dots on it. The ringR(; ) inherits a grading from R(). These quotient rings should be theanalogues of the Ariki-Koike cyclotomic Hecke algebras in our framework.Let

R(; )def=

[I]

R(; ),

and switch from to a field k. We expect that, for sufficiently nice and k, the category of graded modules over R(; ) categorifies the inte-grable irreducible Uq(g)-representation V with the highest weight . LetR(; )pmod be the category of finitely-generated graded projective left

R(; )-modules and

R(; )pmoddef=

[I]

R(; )pmod.

There should exist an isomorphism

K0(R(; )) = V ,,

where V , is an integral version of V, a free [q, q

1]-module spanned by

the compositions of divided differences F(a)

i applied to the highest weight

vector v V. Under this isomorphism indecomposable projectives shouldcorrespond to canonical basis vectors in V. The action of E(a)i and F

(a)i

should lift to exact functors E(a)i and F

(a)i between categories R(; )pmod

and R( + ai; )pmod as well as the categories R(; )mod and R( +

ai; )mod of all finitely-generated graded modules. These functors E(a)i

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REFERENCES 49

and F(a)i will be direct summands of the induction and restriction functors

between R(; ) and R( + ai; )-modules, defined a la Ariki. We expectthem to be biadjoint, up to grading shifts.

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M.K.: School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540,and Department of Mathematics, Columbia University, New York, NY 10027

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A.L: Department of Mathematics, Columbia University, New York, NY 10027email: [email protected]
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a diagrammatic approach to categorifying quantum groups i

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