garnir modules, hall-littlewood polynomials, and ... · garnir modules, hall-littlewood...

Outline Sn Irreducibles Hall-Littlewoods Springer cohomology Connections Conclusion

Garnir modules, Hall-Littlewood polynomials, andcohomology of Springer fibers

Alexander Woo

January 25, 2009

Alexander Woo

Garnir modules, Hall-Littlewood polynomials, and cohomology of Springer fibers


A Construction of Sn Irreducible RepresentationsPreliminariesThe construction

Hall-Littlewood polynomialsThe Frobenius characterHall-Littlewood polynomials

Equivariant cohomology of Springer fibersBackgroundThe Goresky-Macpherson construction

Connections among the aboveThe desired connectionThe map

ConclusionConclusion

Alexander Woo



Preliminaries

Partitions

Let n be a positive integer (fixed throughout the talk).

A partition λ of n is a sequence

λ1 ≥ · · · ≥ λ` > 0

withλ1 + · · ·+ λ` = n.

The number ` is call the length of the partition and usuallywritten as `(λ).

Alexander Woo



Preliminaries

Dominance order

There is a partial order on partitions of n called dominance order

We write λ ≥ µ ifk∑

i=1

λi ≥k∑

i=1

µi

for all k . (Here, λi = 0 by convention if i > `(λ).)

Alexander Woo



Preliminaries

Representation Theory in Two Minutes

A representation of Sn is a vector space V with a linear action ofSn on V .

Given a representation V , a subrepresentation is a subspaceW ⊆ V which is closed under the action of Sn.

If a representation V has no subrepresentations other than {0} andV , it is irreducible.

If we are talking about vector spaces over C (or some other field ofcharacteristic 0), then every representation is a direct sum ofirreducible representations.

Alexander Woo



Preliminaries

Counting Sn irreducible representations

Character theory tells us in general that the number of irreduciblerepresentations of a group is the same as the number of conjugacyclasses.The conjugacy classes of Sn are cycle types.Hence there is one irreducible representation of Sn for eachpartition λ of n.

Alexander Woo



The construction

Young diagrams

Given a partition, we can draw a diagram called a Young diagramby drawing λ1 boxes on the first row, λ2 boxes on the second row,and so on.

Example

n = 8, λ = (4, 3, 1)

λ =

Alexander Woo



The construction

Young diagrams

Given a partition, we can draw a diagram called a Young diagramby drawing λ1 boxes on the first row, λ2 boxes on the second row,and so on.

Example

n = 8, λ = (4, 3, 1), µ = (3, 3, 2)

λ = µ =

A partition λ > µ if we can start with the diagram for λ and moveboxes up to get the diagram for µ.

Alexander Woo



The construction

Fillings

A filling of a Young diagram is a way of putting the numbers1, . . . , n in the boxes, one in each box.

Example

n = 8, λ = (4, 3, 1)

T :=26 3 78 1 4 5

Alexander Woo



The construction

Garnir polynomials

Given a filling T , we can associate a polynomial gT by

gT =∏

i directly under j in T

(xi − xj).

Example

n = 8, λ = (4, 3, 1)

T :=26 3 78 1 4 5

gT = (x8 − x6)(x8 − x2)(x6 − x2)(x1 − x3)(x4 − x7)

Alexander Woo



The construction

The irreducible representation

Let Gλ be the span of {gT | T is a filling of λ}.

Theorem (Garnir?, Young?)

The subspace Gλ ∈ C[x] is an irreducible representation of Sn. Thevector spaces Gλ and Gµ are not isomorphic as representations ifλ 6= µ, so this construction gives all the irreducible representationsof Sn.

Note that the polynomials gT are not linearly independent. Onebasis for Gλ is given by the standard tableaux, which are fillingswhere the number increases as you go along each row and column.

Alexander Woo



The construction

Garnir ideals

So far, we have only added polynomials. But a polynomial can bemultiplied by another!Let Iλ be the ideal in C[x1, . . . , xn] generated by the Garnirpolynomials gT for all fillings T of λ (or equivalently by all thepolynomials in Gλ). We call this ideal a Garnir ideal.

Here is a sign this is something nice:

Proposition

For every λ ≤ µ, every subrepresentation of C[x1, . . . , xn]isomorphic to Gλ is contained in Iµ. In particular, if λ ≤ µ, Iλ ⊆ Iµ.

Alexander Woo



The construction

Garnir modules

Natural question:How does Iλ decompose into a direct sum of irreduciblerepresentations?

It turns out to be easier to work with Garnir modules, defined by

Mµ =Iµ∑

ν<µ Iν.

As Sn representations,

Iλ ∼=⊕µ≤λ

Mµ.

Alexander Woo



The Frobenius character


To simplify notation, it is customary to define a function

F : Sn representations → symmetric polynomials (in z1, . . . , zn)

by

F(Gλ) = sλ

and

F(V ⊕W ) = F(V ) + F(W ).

Alexander Woo




The graded Frobenius character

If V is graded, meaning that we have a given decomposition

V =⊕d∈Z

Vd ,

let

F(V ; t) =∑d∈Z

F(Vd)td .

Alexander Woo




Frobenius character for Garnir modules

A few calculations led to the conjecture that

F(Mµ; t) =Hµ

bµ.

Hµ is the Hall-Littlewood polynomial (next slide).

bµ =∏i

(1− t)(1− t2) · · · (1− tmi (µ)),

where mi (µ) is the number of times i occurs in µ.If µ = (4, 3, 3, 1), then

b(4,3,3,1) = (1− t)(1− t)(1− t2)(1− t).

Alexander Woo



Hall-Littlewood polynomials


TheoremThere exist uniquely symmetric functions H̃µ(t) (in z variables overthe coefficient field C(t)) such that

1. H̃µ(t) is in the span of sλ for λ ≥ µ

2. H̃µ[Z (1− t)](t) is in the span of sλ for λ ≤ µ.

3. H̃µ(0) = s(n) = hn

Alexander Woo








3. H̃µ(0) = s(n) = hn

In terms of Frobenius characteristic, if f = F(V ), then

f [Z (1− t)] =∑d

(−1)d tdF(V ⊗ ∧dCn).

Alexander Woo








3. H̃µ(0) = s(n) = hn

Also,Hµ(t) = tn(µ)H̃µ(t−1).

The power n(µ) =∑

i (i − 1)µi is the degree in t of H̃µ, somultiplying by tn(µ) is what is required to make the lowest t-degreeterm of Hµ(t) the constant term.

Alexander Woo



Background

What kind of thing is equivariant cohomology of Springerfibers?

A Springer fiber is a geometric object. There is one for eachpartition µ, and we denote it Zµ. The torus T = (C∗)`(µ) acts onit.

Cohomology associates a graded ring to any geometric object.We will work with C-coefficients, so our ring will be a C-algebra.The cohomology of Zµ is denoted H∗(Zµ).

Equivariant cohomology associates a graded ring to anygeometric object with an action of some group on it. In this casewhere our group is T , the ring is a C[t1, . . . , t`(µ)] = C[t]-algebra.The equivariant cohomology of Zµ is denoted H∗

T (Zµ).

Alexander Woo



Background

Why did people start studying this?

The cohomology of Springer fibers first attracted attention becausethere is an Sn action on H∗(Zµ) so that the top degree piece is theirreducible representation (isomorphic to) Gµ. This gives ageometric way to construct the irreducible representations.Variants of this theory work in positive characteristic and also tellus about representations of matrix groups over finite fields.

Alexander Woo



Background

Springer fibers and Hall-Littlewood polynomials

We are interested because

F(H∗(Zµ)) = H̃µ(t).

This provided the first proof that, when H̃µ(t) is written in termsof the basis {sλ} of Schur functions, the coefficients arepolynomials with positive integer coefficients.

Alexander Woo



Background

Equivariant formality

As is the case for many spaces, Zµ is equivariantly formal,meaning that

H∗T (Zµ) ∼= H∗(Zµ)⊗C C[t1, . . . , t`(µ)],

as C[t] = C[t1, . . . , t`(µ)] modules, so

F(H∗T (Zµ)) =

H̃µ(t)

(1− t)`(µ)

with Sn acting trivially on the t variables.

Alexander Woo



The Goresky-Macpherson construction

A little algebraic geometry

Given a set of points W ⊆ Ck , the ideal I (W ) is the set ofpolynomials

I (X ) = {f ∈ C[x1, . . . , xk ] | f (x) = 0∀x ∈ W }.

This really is an ideal in the ring theory sense. Usually we assumethat W is defined by polynomials in the first place.

The coordinate ring of W is the ring

C[x1, . . . , xk ]/I (W ).

This is the ring of all polynomial functions W → C, since adding apolynomial in I (W ) does not change the function.

Alexander Woo




A collection of subspaces

Given a partition µ of n, define a subset Wµ ⊆ Cn+`(µ) as follows.

Let Cn+`(µ) have coordinates x1, . . . , xn, t1, . . . , t`(µ).

Alexander Woo






For a filling T of the Young diagram of µ, let WT be the`(µ)-dimensional subspace defined by the equations xi = tjwhenever the number i is in the j-th row.

Example

n = 8, λ = (4, 3, 1)

T :=26 3 78 1 4 5

The subspace WT is defined by the equations x2 = t3,x3 = x6 = x7 = t2, and x1 = x4 = x5 = x8 = t1.

Alexander Woo








Let Wµ be the union of the WT for all fillings T of the Youngdiagram of µ.

Alexander Woo








Let Wµ be the union of the WT for all fillings T of the Youngdiagram of µ.

Theorem (Goresky and Macpherson)

The equivariant cohomology ring H∗T (Zµ) is the coordinate ring of

Wµ.

Alexander Woo



The desired connection

The desired isomorphism

Recall thatF(Mµ) = Hµ(t)/bµ(t)

(conjecturally so far in the talk) and

F(H∗T (Zµ)) = H̃µ(t)/(1− t)`(µ).

Alexander Woo







F(H∗T (Zµ)) = H̃µ(t)/(1− t)`(µ).

To account for the difference between (1− t)`(µ) andbµ(t) =

∏i (1− t) · · · (1− tmi (µ)):

Let Y be the subgroup of S`(µ) generated by the action ofswitching the variables ti and tj if µi = µj .

Then the Y -invariants of H∗T (Zµ) have Frobenius series

F(H∗T (Zµ)Y ) = H̃µ/bµ.

Alexander Woo







F(H∗T (Zµ)) = H̃µ(t)/(1− t)`(µ).

To account for the difference between H̃µ(t) and Hµ(t), we useHom to take a dual along with a grading shift (which we supress inthe notation).

Alexander Woo







F(H∗T (Zµ)) = H̃µ(t)/(1− t)`(µ).

Let Y be the subgroup of S`(µ) generated by the action ofswitching the variables ti and tj if µi = µj .

So we want to show

Mµ∼= HomC[t]Y (H∗

T (Zµ)Y , C[t]Y ).

Alexander Woo



The map

A trace map

Define a maptr : H∗

T (Zµ) → C[t]

by

tr(f ) =∑T

1

n!f |WT

On WT , a polynomial can be just thought of as a polynomial inthe t variables.

Warning: tr is not a ring homomorphism, but it is C[t]-linear.

This map also has a topological meaning.

Alexander Woo



The map

The trace map algebraically, I

Let sym denote the symmetrizing operator

sym(f ) =1

n!

∑σ∈Sn

σf ,

where σ is permuting the x variables only.The trace map commutes with sym; i.e.

tr(sym(f )) = sym(tr(f )),

since permuting the x variables just permutes the subspaces WT .So we understand tr if we understand tr(f ) when f is symmetric inthe x variables.

Alexander Woo



The map

The trace map algebraically, II

Proposition (Garsia and Procesi)

In the coordinate ring of Wµ (and hence on every WT ), if f is asymmetric function in the x variables, then

f (x1, . . . , xn) ∼= f (t1, . . . , t1, t2, . . . , t2, . . . , t`(µ), . . . , t`(µ)),

where ti appears µi times.

So tr(f ) can be calculated by first symmetrizing f (in the xvariables), then setting xj to ti according to some filling T of µ.

Alexander Woo



The map

The trace map on Garnir ideals

LemmaIf f ∈ Iµ is symmetric, then f ∈ I 2

µ .

This follows from Schur’s Lemma, Sn representations being selfdual (having real characters), and the appearance of everyV λ ∈ C[x] in Iµ.

For f ∈ Iµ, symmetrizing f gives a symmetric polynomial in Iµ, andhence a polynomial in I 2

µ .

Substituting ti for xj according to the filling T sends gT to∏i<j(ti − tj)

µj .

It follows that tr(f ) is divisible by∏

i<j(ti − tj)2µj for every f ∈ Iµ.

Alexander Woo



The map

The desired isomorphism map

Letφ : Iµ → HomC[t](H

∗T (Zµ), C[t])

be defined by

φ(f ) = (g 7→ tr(fg)∏i<j(ti − tj)2µj

).

Alexander Woo



The map


The kernel of φ is ∑ν<µ

Iν .

and φ(f ) will always send Y -invariant polynomials to Y -invariantpolynomials, so φ induces an injection

Mµ → HomC[t]Y (H∗T (Zµ)Y , C[t]Y ).

We show this is an isomorphism by comparing the dimension (asC-vector spaces of each degree piece. This requires matching arecurrence of Garsia and Procesi (for the dimensions of H∗(Zµ)) toa recurrence we prove combinatorially for Mµ.

Alexander Woo



Conclusion

Main theorem

I have sketched a proof that

F(Mµ) =Hµ(t)

bµ(t).

As a corollary,

F(Iµ) =∑λ≤µ

Hµ(t)

bµ(t).

Alexander Woo



Conclusion

Where this came from

The original motivation for this came from the Hilbert scheme of npoints on C2. We also prove that C[x]/Iµ are the global sections ofthe Procesi n! bundle over the subscheme corresponding to idealswith particular initial ideals. This would be a topic for a wholeother talk.

The construction of H∗T (Zµ) as the coordinate ring of a variety

(and, in fact, always a subspace arrangement) is part of a generaltheory of Goresky, Kottwicz, and Macpherson.

Alexander Woo



Conclusion

A related problem

Springer fibers and Hall-Littlewood polynomials can be constructedfor other Weyl groups, though in the non-simply-laced case thesituation is less nice; one can get in the top degree piece ofcohomology several copies of a single irreducible representation. Itmight be possible to extend the results of this talk similarly.

Alexander Woo



Conclusion

The End

Thank you for your attention!

Alexander Woo


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