Partition Identities and Quiver Representationsweigandt/QuiverPart.pdf · 2020. 5. 19. ·...

Partition Identities and Quiver Representations

Anna Weigandt

University of Illinois at Urbana-Champaign

weigndt2@illinois.edu

January 23rd, 2017

Based on joint work with Richard Rimanyi and Alexander Yong

arXiv:1608.02030

Anna Weigandt Partition Identities and Quiver Representations

Overview

This project provides an elementary explanation for a quantumdilogarithm identity due to M. Reineke.

We use generating function techniques to establish a relatedidentity, which is a generalization of the Euler-Gauss identity.

This reduces to an equivalent form of Reineke’s identity in type A.

Representations of Quivers

Quivers

A quiver Q = (Q0,Q1) is a directed graph with

vertices: i ∈ Q0

edges: a : i → j ∈ Q1

See notes of Brion ([Bri08])

The Definition

A representation of Q is an assignment of a:

vector space Vi to each vertex i ∈ Q0 and

linear transformation fa : Vi → Vj to each arrow ia−→ j ∈ Q1

dim(V ) = (dimVi )i∈Q0 is the dimension vector of V .

The Definition

A representation of Q is an assignment of a:

vector space Vi to each vertex i ∈ Q0 and

linear transformation fa : Vi → Vj to each arrow ia−→ j ∈ Q1

dim(V ) = (dimVi )i∈Q0 is the dimension vector of V .

Morphisms of Representations

A morphism of quiver representations h : (V , f )→ (W , g) is acollection of linear transformations {hi : i ∈ Q0} so that for everyarrow i

a−→ j :

Two representations are isomorphic if each hi is invertible.

The Representation Space

Fix d ∈ NQ0 , The representation space is

RepQ(d) :=⊕

ia−→j∈Q1

Mat(d(i),d(j)).

LetGLQ(d) :=

∏i∈Q0

GL(d(i)).

GLQ(d) acts on RepQ(d) by base change at each vertex.

Orbits of this action are in bijection with isomorphism classes of ddimensional representations.

Example 1

What parameterizes isomorphism classes?

(g1, g2) · (A) = g1Ag−12

A choice of dimension vector and the rank of the map.

Example 1

(g1, g2) · (A) = g1Ag−12

Example 1

(g1, g2) · (A) = g1Ag−12

Example 1

(g1, g2) · (A) = g1Ag−12

Example 2

(g) · (A) = gAg−1

Jordan form (up to rearranging blocks)

Example 2

(g) · (A) = gAg−1

Example 2

(g) · (A) = gAg−1

Simple Representations

Assume Q has no oriented cycles. The simple representations αi

are formed by assigning C to vertex i and 0 to all other vertices.

Indecomposable Representations

From V and W we can build a new representation V ⊕W .

V is indecomposable if writing V ∼= V ′ ⊕ V ′′ implies V ′ or V ′′ istrivial.

Indecomposable representations are the building blocks for allrepresentations.

Indecomposable Representations

From V and W we can build a new representation V ⊕W .

V is indecomposable if writing V ∼= V ′ ⊕ V ′′ implies V ′ or V ′′ istrivial.

Indecomposable representations are the building blocks for allrepresentations.

Dynkin Quivers

A quiver is Dynkin if its underlying graph is of type ADE :

Gabriel’s Theorem

Theorem ([Gab75])

Dynkin quivers have finitely many isomorphism classes ofindecomposable representations.

For type A, indecomposables V[i ,j] are indexed by intervals.

Gabriel’s Theorem

Theorem ([Gab75])

Dynkin quivers have finitely many isomorphism classes ofindecomposable representations.

For type A, indecomposables V[i ,j] are indexed by intervals.

Reineke’s Identities

The Quantum Dilogarithm Series

E(z) =∞∑k=0

qk2/2zk

(1− q)(1− q2) . . . (1− qk)

This is a generalization of the Rogers dilogarithm

The Quantum Dilogarithm Series

E(z) =∞∑k=0

qk2/2zk

(1− q)(1− q2) . . . (1− qk)

This is a generalization of the Rogers dilogarithm

The Quantum Exponential

Define the quantum exponential:

expq(z) =∞∑k=0

[k]q!,

[k]q! = 1(1 + q)(1 + q + q2) · · · (1 + q + . . .+ qk−1)

is the q-factorial.

The quantum dilogarithm series is:

E(z) = expq

1− qz

The Quantum Exponential

Define the quantum exponential:

expq(z) =∞∑k=0

[k]q!,

[k]q! = 1(1 + q)(1 + q + q2) · · · (1 + q + . . .+ qk−1)

is the q-factorial.

The quantum dilogarithm series is:

E(z) = expq

1− qz

The Pentagon Identity

Theorem (Schutzenberger, Faddeev-Volkov, Faddeev-Kashaev)

If x and y are q-commuting variables, i.e. yx = qxy, then:

E(x)E(y) = E(y)E(q−1/2xy)E(x)

This generalizes the classical pentagon identity.

Reineke extended this idea to define a family of identitiescorresponding to Dynkin quivers.

The Quantum Algebra of a Quiver

Define the Euler form

χ : NQ0 × NQ0 → Z

χ(d1, d2) =∑i∈Q0

d1(i)d2(i) −∑

ia−→j∈Q1

d1(i)d2(j)

The Quantum Algebra of a Quiver

AQ is an algebra over Q(q1/2) with

generators:{yd : d ∈ NQ0}

multiplication:

yd1yd2 = q12λ(d1,d2)yd1+d2

whereλ(d1, d2) = χ(d2, d1)− χ(d1, d2)

Reineke’s Identity

Given a representation V , we’ll write dV as a shorthand for ddim(V ).

For a Dynkin quiver, it is possible to fix a choice of ordering on the

simple representations: α1, . . . , αn

indecomposable representations: β1, . . . , βN

so that

E(yα1) · · ·E(yαn) = E(yβ1) · · ·E(yβN) (1)

(Original proof given by [Rei10], see [Kel11] for exposition and asketch of the proof.)

Reformulation

Looking at the coefficient of yd on each side, this is equivalent tothe following infinite family of identities ([Rim13]):

n∏i=1

(q)d(i)=∑η

qcodimC(η)N∏i=1

(q)mβi(η)

where the sum is over orbits η in RepQ(d) and mβ(η) is themultiplicity of β in V ∈ η.

Here, (q)k = (1− q) . . . (1− qk) is the q-shifted factorial.

Lacing Diagrams

A lacing diagram ([ADF85]) L is a graph so that:

the vertices are arranged in n columns labeled 1, 2, . . . , n

the edges between adjacent columns form a partial matching.

The Role of Lacing Diagrams in Representation Theory

When Q is a type A quiver, a lacing diagram can be interpreted asa sequence of partial permutation matrices which form arepresentation VL of Q.

[ 1 0 0 00 1 0 0

0 0 00 0 01 0 00 1 0

1 00 00 0

See [KMS06] for the equiorientated case and [BR04] forarbitrary orientations.

The Dimension Vector

dim(L) records the number of vertices in each column of L. Notethat dim(L) = dim(VL).

In this lacing diagram,

dim(L) = (2, 4, 3, 2)

The Dimension Vector

dim(L) records the number of vertices in each column of L. Notethat dim(L) = dim(VL).

In this lacing diagram,

dim(L) = (2, 4, 3, 2)

Equivalence Classes of Lacing Diagrams

Two lacing diagrams are equivalent if one can be obtained fromthe other by permuting vertices within a column.

{Equivalence Classes of Lacing Diagrams}

{Isomorphism Classes of Representations of Q}

Equivalence Classes of Lacing Diagrams

Two lacing diagrams are equivalent if one can be obtained fromthe other by permuting vertices within a column.

{Equivalence Classes of Lacing Diagrams}

{Isomorphism Classes of Representations of Q}

Strands

A strand is a connected component of L.

m[i ,j](L) = |{strands starting at column i and ending at column j}|

Example: m[1,2](L) = 2

Strands

Strands record the decomposition of VL into indecomposablerepresentations:

VL ∼= ⊕V⊕m[i,j](L)[i ,j]

Example: VL = V⊕2[1,2] ⊕ V[2,3] ⊕ V[2,4] ⊕ V[3,3] ⊕ V[4,4]

Statistics on Lacing Diagrams

tki (L) = |{strands starting at column i which use column k}|

Example: t32 (L) = 2

ski (L) = m[i ,k−1](L)

Example: s42 (L) = 1

ski (L) = m[i ,k−1](L)

Example: s42 (L) = 1

The Durfee Statistic

Let L(d) = {[L] : dim(L) = d}.

Fix a sequence of permutations w = (w (1), . . . ,w (n)), so thatw (i) ∈ Si and w (i)(i) = i .

Define the Durfee statistic:

rw(L) =∑

1≤i<j≤k≤n

skw (k)(i)

(L)tkw (k)(j)

The Durfee Statistic

Let L(d) = {[L] : dim(L) = d}.

Fix a sequence of permutations w = (w (1), . . . ,w (n)), so thatw (i) ∈ Si and w (i)(i) = i .

Define the Durfee statistic:

rw(L) =∑

1≤i<j≤k≤n

skw (k)(i)

(L)tkw (k)(j)

Theorem (Rimanyi, Weigandt, Yong, 2016)

Fix a dimension vector d = (d(1), . . . ,d(n)) and let w be asbefore. Then

n∏k=1

(q)d(k)=∑

η∈L(d)

qrw(η)n∏

(q)tkk (η)

k−1∏i=1

[tki (η) + ski (η)

ski (η)

Note:[j+k

]qis the q-binomial coefficient.

Generating Series for Partitions

Generating Series

Let S be a set equipped with a weight function

wt : S → N

so that|{s ∈ S : wt(s) = k}| <∞

for each k ∈ N.

The generating series for S is

G (S , q) =∑s∈S

qwt(s).

Partitions

An integer partition is an ordered list of decreasing integers:

λ = λ1 ≥ λ2 ≥ . . . ≥ λℓ(λ) ≥ 0

We will typically represent a partition by its Young diagram:

The Generating Series for Rectangles of Width k

∅ . . .

1 + qk + q2k + q3k + q4k + q5k + . . . =1

1− qk

∅ . . .

1 + qk + q2k + q3k + q4k + q5k + . . . =1

1− qk

∅ . . .

1 + qk + q2k + q3k + q4k + q5k + . . .

1− qk

∅ . . .

1 + qk + q2k + q3k + q4k + q5k + . . . =1

1− qk

(q)∞=

∞∏k=1

(1− qk)=

∞∏k=1

(1 + qk + q2k + q3k + . . .)