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Partition Identities and Quiver Representations
Anna Weigandt
University of Illinois at Urbana-Champaign
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
Anna Weigandt Partition Identities and Quiver Representations
Quivers
A quiver Q = (Q0,Q1) is a directed graph withvertices: i ∈ Q0
edges: a : i→ j ∈ Q1
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
Dynkin Quivers
A quiver is Dynkin if its underlying graph is of type ADE:
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
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.
1234
[ 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.
Anna Weigandt Partition Identities and Quiver Representations
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)
Anna Weigandt Partition Identities and Quiver Representations
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)
Anna Weigandt Partition Identities and Quiver Representations
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
Anna Weigandt Partition Identities and Quiver Representations
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
Anna Weigandt Partition Identities and Quiver Representations
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
Anna Weigandt Partition Identities and Quiver Representations
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[4,4](L) = 1
Anna Weigandt Partition Identities and Quiver Representations
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]
Anna Weigandt Partition Identities and Quiver Representations
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}
Anna Weigandt Partition Identities and Quiver Representations
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}
Anna Weigandt Partition Identities and Quiver Representations
Reineke’s Identities
Anna Weigandt Partition Identities and Quiver Representations
Quantum Dilogarithm Series
E(z) =∞∑
k=0
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.
Anna Weigandt Partition Identities and Quiver Representations
Quantum Dilogarithm Series
E(z) =∞∑
k=0
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.
Anna Weigandt Partition Identities and Quiver Representations
Quantum Dilogarithm Series
E(z) =∞∑
k=0
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.
Anna Weigandt Partition Identities and Quiver Representations
Quantum Dilogarithm Series
E(z) =∞∑
k=0
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.
Anna Weigandt Partition Identities and Quiver Representations
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)
Anna Weigandt Partition Identities and Quiver Representations
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
) (1)
(Original proof given by [Rei10], see [Kel11] for exposition and asketch of the proof.)
Anna Weigandt Partition Identities and Quiver Representations
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∏
i=1
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.
Anna Weigandt Partition Identities and Quiver Representations
A Related Identity (in Type A)
Anna Weigandt Partition Identities and Quiver Representations
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).
Anna Weigandt Partition Identities and Quiver Representations
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).
Anna Weigandt Partition Identities and Quiver Representations
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(η)
k−1∏i=1
[tki (η) + sk
i (η)
ski (η)
]q
[j+kk]
q =(q)j+k(q)j(q)k
is the q-binomial coefficient.
Anna Weigandt Partition Identities and Quiver Representations
Proof Idea
↕
Anna Weigandt Partition Identities and Quiver Representations
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
1
[s21 + t2
1s21
]q
[s31 + t3
1s31
]q
[s41 + t4
1s41
]q
=
(1
(q)t11
)((q)s2
1+t21
(q)s21(q)t2
1
)((q)s3
1+t31
(q)s31(q)t3
1
)((q)s4
1+t41
(q)s41(q)t4
1
)
=1
(q)s21(q)s3
1(q)s4
1(q)t4
1
=1
(q)m[1,1](q)m[1,2](q)m[1,3](q)m[1,4]
Anna Weigandt Partition Identities and Quiver Representations
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
1
[s21 + t2
1s21
]q
[s31 + t3
1s31
]q
[s41 + t4
1s41
]q
=
(1
(q)t11
)((q)s2
1+t21
(q)s21(q)t2
1
)((q)s3
1+t31
(q)s31(q)t3
1
)((q)s4
1+t41
(q)s41(q)t4
1
)
=1
(q)s21(q)s3
1(q)s4
1(q)t4
1
=1
(q)m[1,1](q)m[1,2](q)m[1,3](q)m[1,4]
Anna Weigandt Partition Identities and Quiver Representations
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
1
[s21 + t2
1s21
]q
[s31 + t3
1s31
]q
[s41 + t4
1s41
]q
=
(1
(q)t11
)((q)s2
1+t21
(q)s21(q)t2
1
)((q)s3
1+t31
(q)s31(q)t3
1
)((q)s4
1+t41
(q)s41(q)t4
1
)
=1
(q)s21(q)s3
1(q)s4
1(q)t4
1
=1
(q)m[1,1](q)m[1,2](q)m[1,3](q)m[1,4]
Anna Weigandt Partition Identities and Quiver Representations
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
1
[s21 + t2
1s21
]q
[s31 + t3
1s31
]q
[s41 + t4
1s41
]q
=
(1
(q)t11
)((q)s2
1+t21
(q)s21(q)t2
1
)((q)s3
1+t31
(q)s31(q)t3
1
)((q)s4
1+t41
(q)s41(q)t4
1
)
=1
(q)s21(q)s3
1(q)s4
1(q)t4
1
=1
(q)m[1,1](q)m[1,2](q)m[1,3](q)m[1,4]
Anna Weigandt Partition Identities and Quiver Representations
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
1
[s21 + t2
1s21
]q
[s31 + t3
1s31
]q
[s41 + t4
1s41
]q
=
(1
(q)t11
)((q)s2
1+t21
(q)s21(q)t2
1
)((q)s3
1+t31
(q)s31(q)t3
1
)((q)s4
1+t41
(q)s41(q)t4
1
)
=1
(q)s21(q)s3
1(q)s4
1(q)t4
1
=1
(q)m[1,1](q)m[1,2](q)m[1,3](q)m[1,4]
Anna Weigandt Partition Identities and Quiver Representations
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
1
[s21 + t2
1s21
]q
[s31 + t3
1s31
]q
[s41 + t4
1s41
]q
=
(1
(q)t11
)((q)s2
1+t21
(q)s21(q)t2
1
)((q)s3
1+t31
(q)s31(q)t3
1
)((q)s4
1+t41
(q)s41(q)t4
1
)
=1
(q)s21(q)s3
1(q)s4
1(q)t4
1
=1
(q)m[1,1](q)m[1,2](q)m[1,3](q)m[1,4]
Anna Weigandt Partition Identities and Quiver Representations
Simplifying the Identity
Doing these cancellations yields the identity:
Corollary (Rimányi, Weigandt, Yong, 2016)n∏
i=1
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∏
i=1
1(q)mβi (η)
.
Anna Weigandt Partition Identities and Quiver Representations
Simplifying the Identity
Doing these cancellations yields the identity:
Corollary (Rimányi, Weigandt, Yong, 2016)n∏
i=1
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∏
i=1
1(q)mβi (η)
.
Anna Weigandt Partition Identities and Quiver Representations
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)
Q )
Example:
1 2 3 4 5 6
a1 a2 a3 a4 a5
wQ = (1, 12, 123, 3214, 32145, 541236)
Anna Weigandt Partition Identities and Quiver Representations
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)
Q )
Example:
1 2 3 4 5 6
a1 a2 a3 a4 a5
wQ = (1, 12, 123, 3214, 32145, 541236)
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
Generating Series for Partitions
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
Two Basic Properties
G(S× T, q) = G(S, q)G(T, q)
G(S ⊔ T, q) = G(S, q) + G(T, q)
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
The Generating Series for Rectangles of Width k
∅ . . .
1 + qk + q2k + q3k + q4k + q5k + . . . =1
1− qk
Anna Weigandt Partition Identities and Quiver Representations
The Generating Series for Rectangles of Width k
∅ . . .
1 + qk + q2k + q3k + q4k + q5k + . . . =1
1− qk
Anna Weigandt Partition Identities and Quiver Representations
The Generating Series for Rectangles of Width k
∅ . . .
1 + qk + q2k + q3k + q4k + q5k + . . .
=1
1− qk
Anna Weigandt Partition Identities and Quiver Representations
The Generating Series for Rectangles of Width k
∅ . . .
1 + qk + q2k + q3k + q4k + q5k + . . . =1
1− qk
Anna Weigandt Partition Identities and Quiver Representations
Generating Series for Partitions
1(q)∞
=∞∏
k=1
1(1− qk)
=∞∏
k=1(1 + qk + q2k + q3k + . . .)
q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .
Anna Weigandt Partition Identities and Quiver Representations
Generating Series for Partitions
1(q)∞
=∞∏
k=1
1(1− qk)
=∞∏
k=1(1 + qk + q2k + q3k + . . .)
q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .
Anna Weigandt Partition Identities and Quiver Representations
Generating Series for Partitions
1(q)∞
=∞∏
k=1
1(1− qk)
=∞∏
k=1(1 + qk + q2k + q3k + . . .)
q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .
Anna Weigandt Partition Identities and Quiver Representations
Generating Series for Partitions
1(q)∞
=∞∏
k=1
1(1− qk)
=∞∏
k=1(1 + qk + q2k + q3k + . . .)
q16 = q1 · 1 · q3·3 · 1 · 1 · q6 · 1 · 1 · . . .
Anna Weigandt Partition Identities and Quiver Representations
Notation
Let R(j, k) be the set consisting of a single rectangularpartition of size j× k.
G(R(j, k), q) =
qjk
Let P(j, k) be the set of partitions constrained to a j× kbox. (Here, we allow j, k =∞).
G(P(j, k), q) = ??
Anna Weigandt Partition Identities and Quiver Representations
Notation
Let R(j, k) be the set consisting of a single rectangularpartition of size j× k.
G(R(j, k), q) = qjk
Let P(j, k) be the set of partitions constrained to a j× kbox. (Here, we allow j, k =∞).
G(P(j, k), q) = ??
Anna Weigandt Partition Identities and Quiver Representations
Notation
Let R(j, k) be the set consisting of a single rectangularpartition of size j× k.
G(R(j, k), q) = qjk
Let P(j, k) be the set of partitions constrained to a j× kbox. (Here, we allow j, k =∞).
G(P(j, k), q) =
??
Anna Weigandt Partition Identities and Quiver Representations
Notation
Let R(j, k) be the set consisting of a single rectangularpartition of size j× k.
G(R(j, k), q) = qjk
Let P(j, k) be the set of partitions constrained to a j× kbox. (Here, we allow j, k =∞).
G(P(j, k), q) = ??
Anna Weigandt Partition Identities and Quiver Representations
P(∞, k): Partitions With at Most k Columns
Idea: Truncate the product
1(q)k
=k∏
i=1
1(1− qi)
Anna Weigandt Partition Identities and Quiver Representations
P(∞, k): Partitions With at Most k Columns
Idea: Truncate the product
1(q)k
=k∏
i=1
1(1− qi)
Anna Weigandt Partition Identities and Quiver Representations
P(k,∞): Partitions With at Most k Rows
Idea: Bijection via conjugation
1(q)k
=k∏
i=1
1(1− qi)
Anna Weigandt Partition Identities and Quiver Representations
P(k,∞): Partitions With at Most k Rows
Idea: Bijection via conjugation
1(q)k
=k∏
i=1
1(1− qi)
Anna Weigandt Partition Identities and Quiver Representations
q-Binomial Coefficients
Ordinary Binomial Coefficient:If x and y commute,
(x + y)n =∑
i+j=n
(i + j
i
)xiyj.
q-Binomial Coefficient:If x and y are q commuting (yx = qxy),
(x + y)n =∑
i+j=n
[i + j
i
]qxiyj.
Anna Weigandt Partition Identities and Quiver Representations
q-Binomial Coefficients
Ordinary Binomial Coefficient:If x and y commute,
(x + y)n =∑
i+j=n
(i + j
i
)xiyj.
q-Binomial Coefficient:If x and y are q commuting (yx = qxy),
(x + y)n =∑
i+j=n
[i + j
i
]qxiyj.
Anna Weigandt Partition Identities and Quiver Representations
Combinatorial Interpretation of the q-Binomial Coefficient
(x + y)11 = . . .+ yyxyxyyxxyy + . . .
Anna Weigandt Partition Identities and Quiver Representations
Combinatorial Interpretation of the q-Binomial Coefficient
(x + y)11 = . . .+ yyxyxyyxxyy + . . .
= . . .+ q15x4y7 + . . .
xy y x
y xy y x
yy
G(P(i, j), q) =[i + j
i
]q
Anna Weigandt Partition Identities and Quiver Representations
Combinatorial Interpretation of the q-Binomial Coefficient
(x + y)11 = . . .+ yyxyxyyxxyy + . . .
= . . .+ q15x4y7 + . . .
xy y x
y xy y x
yy
G(P(i, j), q) =[i + j
i
]q
Anna Weigandt Partition Identities and Quiver Representations
Combinatorial Interpretation of the q-Binomial Coefficient
(x + y)11 = . . .+ yyxyxyyxxyy + . . .
= . . .+ q15x4y7 + . . .
xy y x
y xy y x
yy
G(P(i, j), q) =[i + j
i
]q=
(q)i+j(q)i(q)j
Anna Weigandt Partition Identities and Quiver Representations
Combinatorial Interpretation of the q-Binomial Coefficient
(x + y)11 = . . .+ yyxyxyyxxyy + . . .
= . . .+ q15x4y7 + . . .
xy y x
y xy y x
yy
G(P(i, j), q) =[i + j
i
]q
=(q)i+j(q)i(q)j
Anna Weigandt Partition Identities and Quiver Representations
Combinatorial Interpretation of the q-Binomial Coefficient
(x + y)11 = . . .+ yyxyxyyxxyy + . . .
= . . .+ q15x4y7 + . . .
xy y x
y xy y x
yy
G(P(i, j), q) =[i + j
i
]q=
(q)i+j(q)i(q)j
Anna Weigandt Partition Identities and Quiver Representations
Durfee Squares and Rectangles
Anna Weigandt Partition Identities and Quiver Representations
Durfee Squares
The Durfee square D(λ) is the largest j× j square partitioncontained in λ.
Anna Weigandt Partition Identities and Quiver Representations
Durfee Squares
The Durfee square D(λ) is the largest j× j square partitioncontained in λ.
Anna Weigandt Partition Identities and Quiver Representations
Durfee Squares
P(∞,∞) ←→∪j≥0R(j, j)× P(j,∞)× P(∞, j)
Euler-Gauss identity:
1(q)∞
=∞∑
j=0
qj2
(q)j(q)j(2)
See [And98] for details and related identities.
Anna Weigandt Partition Identities and Quiver Representations
Durfee Squares
P(∞,∞) ←→∪j≥0R(j, j)× P(j,∞)× P(∞, j)
Euler-Gauss identity:
1(q)∞
=∞∑
j=0
qj2
(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)∞
=∞∑
s=0
qs(s+r)
(q)s(q)s+r. (3)
([GH68])
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)∞
=∞∑
s=0
qs(s+r)
(q)s(q)s+r. (3)
([GH68])
Anna Weigandt Partition Identities and Quiver Representations
Proof of the Main Theorem
Anna Weigandt Partition Identities and Quiver Representations
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(η)
k−1∏i=1
[tki (η) + sk
i (η)
ski (η)
]q
Idea: We will interpret each side as a generating series for tuples ofpartitions. Giving a weight preserving bijection between these twosets proves the identity.
Anna Weigandt Partition Identities and Quiver Representations
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(η)
k−1∏i=1
[tki (η) + sk
i (η)
ski (η)
]q
Idea: We will interpret each side as a generating series for tuples ofpartitions. Giving a weight preserving bijection between these twosets proves the identity.
Anna Weigandt Partition Identities and Quiver Representations
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)
Anna Weigandt Partition Identities and Quiver Representations
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)
Anna Weigandt Partition Identities and Quiver Representations
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)
Anna Weigandt Partition Identities and Quiver Representations
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)
Anna Weigandt Partition Identities and Quiver Representations
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)
Anna Weigandt Partition Identities and Quiver Representations
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)
Anna Weigandt Partition Identities and Quiver Representations
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∏
k=1
1(q)tk
k(η)
k−1∏i=1
[tki (η) + sk
i (η)
ski (η)
]q
Anna Weigandt Partition Identities and Quiver Representations
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∏
k=1
1(q)tk
k(η)
k−1∏i=1
[tki (η) + sk
i (η)
ski (η)
]q
Anna Weigandt Partition Identities and Quiver Representations
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
Anna Weigandt Partition Identities and Quiver Representations
λ(1)
s21
t11
Anna Weigandt Partition Identities and Quiver Representations
λ(1)
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(2)
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(2)
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(2)
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(2)
t21
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(2)
t21t2
2
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(3)
t21t2
2
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3)
t21t2
2
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3)
t21t2
2
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3)
t31
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3)
t31t3
2t33
s31
s32
Anna Weigandt Partition Identities and Quiver Representations
λ(4)
t31t3
2t33
Anna Weigandt Partition Identities and Quiver Representations
λ(4)
t31t3
2t33
Anna Weigandt Partition Identities and Quiver Representations
λ(4)
t31t3
2t33
Anna Weigandt Partition Identities and Quiver Representations
λ(4)
t41
s41
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
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
1t22
s21
t31t3
2t33
s31
s32
t41t4
2t43t4
4s41
s42
s43
λ 7→ (µ,ν) ∈ T(η) ⊆ T
Anna Weigandt Partition Identities and Quiver Representations
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
1t22
s21
t31t3
2t33
s31
s32
t41t4
2t43t4
4s41
s42
s43
λ 7→ (µ,ν) ∈ T(η) ⊆ T
Anna Weigandt Partition Identities and Quiver Representations
λ 7→ (µ,ν)
∅
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
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.
Anna Weigandt Partition Identities and Quiver Representations
Merci!
Anna Weigandt Partition Identities and Quiver Representations
Anna Weigandt Partition Identities and Quiver Representations
What happens with a different choice of w?
Anna Weigandt Partition Identities and Quiver Representations
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
Anna Weigandt Partition Identities and Quiver Representations
λ(1) w(1) = 1
s21
t11
Anna Weigandt Partition Identities and Quiver Representations
λ(1) w(1) = 1
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(2) w(2) = 12
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(2) w(2) = 12
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(2) w(2) = 12
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(2) w(2) = 12
t21
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(2) w(2) = 12
t21t2
2
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(3) w(3) = 213
t21 t2
2
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3) w(3) = 213
t21 t2
2
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3) w(3) = 213
t21 t2
2
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3) w(3) = 213
t31
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3) w(3) = 213
t32t3
1t33
s32
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(4) w(4) = 3124
t31t3
2 t33
Anna Weigandt Partition Identities and Quiver Representations
λ(4) w(4) = 3124
t31t3
2 t33
Anna Weigandt Partition Identities and Quiver Representations
λ(4) w(4) = 3124
t31t3
2 t33
Anna Weigandt Partition Identities and Quiver Representations
λ(4) w(4) = 3124
t41
s41
Anna Weigandt Partition Identities and Quiver Representations
λ(4) w(4) = 3124
t41t4
2 t43t4
4
s41
s42
s43
Anna Weigandt Partition Identities and Quiver Representations
The Associated Lacing Diagram
versus
Anna Weigandt Partition Identities and Quiver Representations
The Associated Lacing Diagram
versus
Anna Weigandt Partition Identities and Quiver Representations