Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5....

Partition Identities and Quiver Representations

Anna Weigandt

University of Illinois at Urbana-Champaign

weigndt2@illinois.edu

September 30th, 2017

Based on joint work with Richárd Rimányi and Alexander Yong

arXiv:1608.02030Anna 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.

Anna Weigandt Partition Identities and Quiver Representations

Representations of Quivers

Quivers

A quiver Q = (Q0,Q1) is a directed graph withvertices: i ∈ Q0

edges: a : i→ j ∈ Q1

The Definition

A representation of Q is an assignment of a:vector space Vi to each vertex i ∈ Q0 andlinear transformation fa : Vi → Vj to each arrow i a−→ j ∈ Q1

dim(V) = (dimVi)i∈Q0 ∈ NQ0 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) :=⊕

i a−→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.

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.

Lacing Diagrams

A lacing diagram ([ADF85]) L is a graph so that:the vertices are arranged in n columns labeled 1, 2, . . . , nthe edges between adjacent columns form a partial matching.

Idea: Lacing diagrams are a way to visually encode representationsof an An quiver.

Lacing Diagrams

A lacing diagram ([ADF85]) L is a graph so that:the vertices are arranged in n columns labeled 1, 2, . . . , nthe edges between adjacent columns form a partial matching.

Idea: Lacing diagrams are a way to visually encode representationsof an An quiver.

Lacing Diagrams and the Representation SpaceWhen 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)

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]

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}

Reineke’s Identities

Quantum Dilogarithm Series

E(z) =∞∑

qk2/2zk

(1− q)(1− q2) . . . (1− qk)

Theorem (Schützenberger, 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 of Rogersdilogarithms.

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

E(z) =∞∑

qk2/2zk

(1− q)(1− q2) . . . (1− qk)

E(z) =∞∑

qk2/2zk

(1− q)(1− q2) . . . (1− qk)

E(z) =∞∑

qk2/2zk

(1− q)(1− q2) . . . (1− qk)

The Quantum Algebra of a Quiver

The Quantum Algebra AQ is an algebra over Q(q1/2) withgenerators:

{yd : d ∈ NQ0}

multiplication:

yd1yd2 = q 12 (χ(d2,d1)−χ(d1,d2))yd1+d2

The Euler form χ : NQ0 × NQ0 → Z

χ(d1,d2) =∑i∈Q0

d1(i)d2(i) −∑

i a−→j∈Q1

d1(i)d2(j)

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 thesimple representations: α1, . . . , αn

indecomposable representations: β1, . . . , βN

so thatE(ydα1

) · · ·E(ydαn ) = E(ydβ1) · · ·E(ydβN

(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

1(q)d(i)

=∑η

qcodimC(η)N∏

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.

A Related Identity (in Type A)

The Durfee Statistic

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

Also fix “bookkeeping statistics”

ski (L) = m[i,k−1](L) and tk

i (L) =n∑

j=km[i,j](L)

Define the Durfee statistic:

rw(L) =∑

1≤i<j≤k≤nskw(k)(i)(L)t

kw(k)(j)(L).

The Durfee Statistic

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

Also fix “bookkeeping statistics”

ski (L) = m[i,k−1](L) and tk

i (L) =n∑

j=km[i,j](L)

Define the Durfee statistic:

rw(L) =∑

1≤i<j≤k≤nskw(k)(i)(L)t

kw(k)(j)(L).

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

Theorem (Rimányi, Weigandt, Yong, 2016)Fix a dimension vector d = (d(1), . . . ,d(n)) and let w be asbefore. Then

n∏k=1

1(q)d(k)

η∈L(d)qrw(η)

n∏k=1

1(q)tk

k−1∏i=1

[tki (η) + sk

i (η)

ski (η)

[j+kk]

q =(q)j+k(q)j(q)k

is the q-binomial coefficient.

Proof Idea

Connection to Reineke’s Identities

Notice that ski + tk

i = tk−1i .

Lets think about n = 4 and just the terms with subscript 1:

1(q)t1

[s21 + t2

[s31 + t3

[s41 + t4

(q)t11

)((q)s2

(q)s21(q)t2

)((q)s3

(q)s31(q)t3

)((q)s4

(q)s41(q)t4

(q)s21(q)s3

1(q)s4

1(q)t4

(q)m[1,1](q)m[1,2](q)m[1,3](q)m[1,4]

i = tk−1i .

1(q)t1

[s21 + t2

[s31 + t3

[s41 + t4

(q)t11

)((q)s2

(q)s21(q)t2

)((q)s3

(q)s31(q)t3

)((q)s4

(q)s41(q)t4

(q)s21(q)s3

1(q)s4

1(q)t4

(q)m[1,1](q)m[1,2](q)m[1,3](q)m[1,4]

i = tk−1i .

1(q)t1

[s21 + t2

[s31 + t3

[s41 + t4

(q)t11

)((q)s2

(q)s21(q)t2

)((q)s3

(q)s31(q)t3

)((q)s4

(q)s41(q)t4

(q)s21(q)s3

1(q)s4

1(q)t4

(q)m[1,1](q)m[1,2](q)m[1,3](q)m[1,4]

i = tk−1i .

1(q)t1

[s21 + t2

[s31 + t3

[s41 + t4

(q)t11

)((q)s2

(q)s21(q)t2

)((q)s3

(q)s31(q)t3

)((q)s4

(q)s41(q)t4

(q)s21(q)s3

1(q)s4

1(q)t4

(q)m[1,1](q)m[1,2](q)m[1,3](q)m[1,4]

i = tk−1i .

1(q)t1

[s21 + t2

[s31 + t3

[s41 + t4

(q)t11

)((q)s2

(q)s21(q)t2

)((q)s3

(q)s31(q)t3

)((q)s4

(q)s41(q)t4

(q)s21(q)s3

1(q)s4

1(q)t4

(q)m[1,1](q)m[1,2](q)m[1,3](q)m[1,4]

i = tk−1i .

1(q)t1

[s21 + t2

[s31 + t3

[s41 + t4

(q)t11

)((q)s2

(q)s21(q)t2

)((q)s3

(q)s31(q)t3

)((q)s4

(q)s41(q)t4

(q)s21(q)s3

1(q)s4

1(q)t4

(q)m[1,1](q)m[1,2](q)m[1,3](q)m[1,4]

Simplifying the Identity

Doing these cancellations yields the identity:

Corollary (Rimányi, Weigandt, Yong, 2016)n∏

1(q)d(i)

η∈L(d)qrw(η)

∏1≤i≤j≤n

1(q)m[i,j](η)

which looks very similar to:

n∏i=1

1(q)d(i)

=∑η

qcodimC(η)N∏

1(q)mβi (η)

Simplifying the Identity

Doing these cancellations yields the identity:

Corollary (Rimányi, Weigandt, Yong, 2016)n∏

1(q)d(i)

η∈L(d)qrw(η)

∏1≤i≤j≤n

1(q)m[i,j](η)

which looks very similar to:

n∏i=1

1(q)d(i)

=∑η

qcodimC(η)N∏

1(q)mβi (η)

A Special Sequence of PermutationsWe associate permutations w(i)

Q ∈ Si to Q as follows:

Let w(1)Q = 1 and w(2)

Q = 12.

If ai−2 and ai−1 point in the same direction, append i to w(i−1)Q

If ai−2 and ai−1 point in opposite directions, reverse w(i−1)Q

and then append i

wQ := (w(1)Q , . . . ,w(n)

Example:

1 2 3 4 5 6

a1 a2 a3 a4 a5

wQ = (1, 12, 123, 3214, 32145, 541236)

A Special Sequence of PermutationsWe associate permutations w(i)

Q ∈ Si to Q as follows:

Let w(1)Q = 1 and w(2)

Q = 12.

If ai−2 and ai−1 point in the same direction, append i to w(i−1)Q

If ai−2 and ai−1 point in opposite directions, reverse w(i−1)Q

and then append i

wQ := (w(1)Q , . . . ,w(n)

Example:

1 2 3 4 5 6

a1 a2 a3 a4 a5

wQ = (1, 12, 123, 3214, 32145, 541236)

The Geometric Meaning of rw(η)

The Durfee statistic has the following geometric meaning:

Theorem (Rimányi, Weigandt, Yong, 2016)

codimC(η) = rwQ(η)

The above statement combined with the corollary implies Reineke’squantum dilogarithm identity in type A.

Generating Series for Partitions

Generating Series

Let S be a set equipped with a weight function

wt : S→ N

so thatSk := |{s ∈ S : wt(s) = k}| <∞

for each k ∈ N.

The generating series for S is

G(S, q) =∑s∈S

qwt(s) =∞∑

k=0Skqk.

Two Basic Properties

G(S× T, q) = G(S, q)G(T, q)

G(S ⊔ T, q) = G(S, q) + G(T, q)

Partitions

An integer partition λ is an ordered list of decreasing integers:

λ1 ≥ λ2 ≥ . . . ≥ λk > 0

We will typically represent a partition by its Young diagram:

We weight a partition by counting the number of boxes it has.

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

1(q)∞

=∞∏

1(1− qk)

=∞∏

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

q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .

1(q)∞

=∞∏

1(1− qk)

=∞∏

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

q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .

1(q)∞

=∞∏

1(1− qk)

=∞∏

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

q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .

1(q)∞

=∞∏

1(1− qk)

=∞∏

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

q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .

Notation

Let R(j, k) be the set consisting of a single rectangularpartition of size j× k.

G(R(j, k), q) =

Let P(j, k) be the set of partitions constrained to a j× kbox. (Here, we allow j, k =∞).

G(P(j, k), q) = ??

Notation

G(R(j, k), q) = qjk

G(P(j, k), q) = ??

Notation

G(R(j, k), q) = qjk

G(P(j, k), q) =

Notation

G(R(j, k), q) = qjk

G(P(j, k), q) = ??

P(∞, k): Partitions With at Most k Columns

Idea: Truncate the product

1(1− qi)

P(∞, k): Partitions With at Most k Columns

Idea: Truncate the product

1(1− qi)

P(k,∞): Partitions With at Most k Rows

Idea: Bijection via conjugation

1(1− qi)

P(k,∞): Partitions With at Most k Rows

Idea: Bijection via conjugation

1(1− qi)

q-Binomial Coefficients

Ordinary Binomial Coefficient:If x and y commute,

(x + y)n =∑

(i + j

)xiyj.

q-Binomial Coefficient:If x and y are q commuting (yx = qxy),

(x + y)n =∑

[i + j

]qxiyj.

q-Binomial Coefficients

Ordinary Binomial Coefficient:If x and y commute,

(x + y)n =∑

(i + j

)xiyj.

q-Binomial Coefficient:If x and y are q commuting (yx = qxy),

(x + y)n =∑

[i + j

]qxiyj.

Combinatorial Interpretation of the q-Binomial Coefficient

(x + y)11 = . . .+ yyxyxyyxxyy + . . .

= . . .+ q15x4y7 + . . .

xy y x

y xy y x

G(P(i, j), q) =[i + j

(x + y)11 = . . .+ yyxyxyyxxyy + . . .

= . . .+ q15x4y7 + . . .

xy y x

y xy y x

G(P(i, j), q) =[i + j

(x + y)11 = . . .+ yyxyxyyxxyy + . . .

= . . .+ q15x4y7 + . . .

xy y x

y xy y x

G(P(i, j), q) =[i + j

(q)i+j(q)i(q)j

(x + y)11 = . . .+ yyxyxyyxxyy + . . .

= . . .+ q15x4y7 + . . .

xy y x

y xy y x

G(P(i, j), q) =[i + j

=(q)i+j(q)i(q)j

(x + y)11 = . . .+ yyxyxyyxxyy + . . .

= . . .+ q15x4y7 + . . .

xy y x

y xy y x

G(P(i, j), q) =[i + j

(q)i+j(q)i(q)j

Durfee Squares and Rectangles

Durfee Squares

The Durfee square D(λ) is the largest j× j square partitioncontained in λ.

Durfee Squares

The Durfee square D(λ) is the largest j× j square partitioncontained in λ.

Durfee Squares

P(∞,∞) ←→∪j≥0R(j, j)× P(j,∞)× P(∞, j)

Euler-Gauss identity:

1(q)∞

=∞∑

(q)j(q)j(2)

See [And98] for details and related identities.

Durfee Squares

P(∞,∞) ←→∪j≥0R(j, j)× P(j,∞)× P(∞, j)

Euler-Gauss identity:

1(q)∞

=∞∑

(q)j(q)j(2)

See [And98] for details and related identities.Anna Weigandt Partition Identities and Quiver Representations

Durfee Rectangles

D(λ,−1) D(λ, 0) D(λ, 4)

The Durfee Rectangle D(λ, r) is the largest s× (s + r)rectangular partition contained in λ.

1(q)∞

=∞∑

qs(s+r)

(q)s(q)s+r. (3)

([GH68])

Durfee Rectangles

D(λ,−1) D(λ, 0) D(λ, 4)

The Durfee Rectangle D(λ, r) is the largest s× (s + r)rectangular partition contained in λ.

1(q)∞

=∞∑

qs(s+r)

(q)s(q)s+r. (3)

([GH68])

Proof of the Main Theorem

n∏k=1

1(q)d(k)

η∈L(d)qrw(η)

n∏k=1

1(q)tk

k−1∏i=1

[tki (η) + sk

i (η)

ski (η)

Idea: We will interpret each side as a generating series for tuples ofpartitions. Giving a weight preserving bijection between these twosets proves the identity.

n∏k=1

1(q)d(k)

η∈L(d)qrw(η)

n∏k=1

1(q)tk

k−1∏i=1

[tki (η) + sk

i (η)

ski (η)

Idea: We will interpret each side as a generating series for tuples ofpartitions. Giving a weight preserving bijection between these twosets proves the identity.

The Left Hand Side

S = P(∞,d(1))× . . .× P(∞,d(n))

G(S, q) =n∏

i=1G(P(∞,d(i)), q) =

n∏i=1

1(q)d(i)

The Left Hand Side

S = P(∞,d(1))× . . .× P(∞,d(n))

G(S, q) =n∏

i=1G(P(∞,d(i)), q) =

n∏i=1

1(q)d(i)

The Right Hand Side

T =∪

η∈L(d)R(η)× P(η)

G(T, q) =∑

η∈L(d)G(R(η), q)G(P(η), q)

The Right Hand Side

T =∪

η∈L(d)R(η)× P(η)

G(T, q) =∑

η∈L(d)G(R(η), q)G(P(η), q)

Rectangles

R(η) = {µ = (µki,j) : µ

ki,j ∈ R(sk

w(k)(i)(η), tkw(k)(j)(η)), 1 ≤ i < j ≤ k ≤ n}

Recall the Durfee Statistic:

rw(η) =∑

1≤i<j≤k≤nskw(k)(i)(η)t

kw(k)(j)(η)

G(R(η), q) = qrw(η)

Rectangles

R(η) = {µ = (µki,j) : µ

ki,j ∈ R(sk

w(k)(i)(η), tkw(k)(j)(η)), 1 ≤ i < j ≤ k ≤ n}

Recall the Durfee Statistic:

rw(η) =∑

1≤i<j≤k≤nskw(k)(i)(η)t

kw(k)(j)(η)

G(R(η), q) = qrw(η)

Partitions with Constrained Size

For convenience, write skk(η) =∞.

P(η) = {ν = (νki ) : ν

ki ∈ P(sk

w(k)(i)(η), tkw(k)(i)(η)), 1 ≤ i ≤ k ≤ n}

G(P(η), q) =n∏

1(q)tk

k−1∏i=1

[tki (η) + sk

i (η)

ski (η)

Partitions with Constrained Size

For convenience, write skk(η) =∞.

P(η) = {ν = (νki ) : ν

ki ∈ P(sk

w(k)(i)(η), tkw(k)(i)(η)), 1 ≤ i ≤ k ≤ n}

G(P(η), q) =n∏

1(q)tk

k−1∏i=1

[tki (η) + sk

i (η)

ski (η)

The map S→ T

Let w = (1, 12, 123, 1234) and d = (8, 9, 11, 8).

λ = (λ(1), λ(2), λ(3), λ(4)) ∈ S

These Parameters are Well Defined

LemmaThere exists a unique η ∈ L(d) so that sk

i (η) = ski and tk

j (η) = tkj

for all i, j, k.

A Recursion

For any η,ski (η) + tk

i (η) = tk−1i (η) (4)

The parameters defined by Durfee rectangles satisfy the sameequations:

t11 t2

2t43t4

λ 7→ (µ,ν) ∈ T(η) ⊆ T

A Recursion

For any η,ski (η) + tk

i (η) = tk−1i (η) (4)

The parameters defined by Durfee rectangles satisfy the sameequations:

t11 t2

2t43t4

λ 7→ (µ,ν) ∈ T(η) ⊆ T

λ 7→ (µ,ν)

References I

S. Abeasis and A. Del Fra.Degenerations for the representations of a quiver of type Am.Journal of Algebra, 93(2):376–412, 1985.

G. E. Andrews.The theory of partitions.2. Cambridge university press, 1998.A. S. Buch and R. Rimányi.A formula for non-equioriented quiver orbits of type a.arXiv preprint math/0412073, 2004.

M. Brion.Representations of quivers.2008.

References II

P. Gabriel.Finite representation type is open.In Representations of algebras, pages 132–155. Springer, 1975.B. Gordon and L. Houten.Notes on plane partitions. ii.Journal of Combinatorial Theory, 4(1):81–99, 1968.

B. Keller.On cluster theory and quantum dilogarithm identities.In Representations of algebras and related topics, EMS Ser.Congr. Rep., pages 85–116. Eur. Math. Soc., Zürich, 2011.A. Knutson, E. Miller and M. Shimozono.Four positive formulae for type A quiver polynomials.Inventiones mathematicae, 166(2):229–325, 2006.

References III

M. Reineke.Poisson automorphisms and quiver moduli.Journal of the Institute of Mathematics of Jussieu,9(03):653–667, 2010.

R. Rimányi.On the cohomological Hall algebra of Dynkin quivers.arXiv:1303.3399, 2013.

Merci!

What happens with a different choice of w?

The map S→ T

Let w = (1, 12, 213, 3124) and d = (8, 9, 11, 8).

λ = (λ(1), λ(2), λ(3), λ(4)) ∈ S

λ(1) w(1) = 1

λ(2) w(2) = 12

λ(3) w(3) = 213

t21 t2

λ(3) w(3) = 213

t21 t2

λ(3) w(3) = 213

t21 t2

λ(3) w(3) = 213

λ(4) w(4) = 3124

2 t43t4

The Associated Lacing Diagram

versus

The Associated Lacing Diagram

versus

Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5....

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