![Page 1: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/1.jpg)
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
![Page 2: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/2.jpg)
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
![Page 3: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/3.jpg)
Representations of Quivers
Anna Weigandt Partition Identities and Quiver Representations
![Page 4: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/4.jpg)
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
![Page 5: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/5.jpg)
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
![Page 6: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/6.jpg)
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
![Page 7: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/7.jpg)
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
![Page 8: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/8.jpg)
Dynkin Quivers
A quiver is Dynkin if its underlying graph is of type ADE:
Anna Weigandt Partition Identities and Quiver Representations
![Page 9: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/9.jpg)
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
![Page 10: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/10.jpg)
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
![Page 11: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/11.jpg)
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
![Page 12: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/12.jpg)
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
![Page 13: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/13.jpg)
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
![Page 14: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/14.jpg)
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
![Page 15: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/15.jpg)
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
![Page 16: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/16.jpg)
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
![Page 17: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/17.jpg)
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
![Page 18: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/18.jpg)
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
![Page 19: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/19.jpg)
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
![Page 20: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/20.jpg)
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
![Page 21: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/21.jpg)
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
![Page 22: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/22.jpg)
Reineke’s Identities
Anna Weigandt Partition Identities and Quiver Representations
![Page 23: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/23.jpg)
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
![Page 24: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/24.jpg)
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
![Page 25: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/25.jpg)
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
![Page 26: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/26.jpg)
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
![Page 27: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/27.jpg)
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
![Page 28: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/28.jpg)
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
![Page 29: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/29.jpg)
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
![Page 30: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/30.jpg)
A Related Identity (in Type A)
Anna Weigandt Partition Identities and Quiver Representations
![Page 31: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/31.jpg)
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
![Page 32: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/32.jpg)
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
![Page 33: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/33.jpg)
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
![Page 34: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/34.jpg)
Proof Idea
↕
Anna Weigandt Partition Identities and Quiver Representations
![Page 35: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/35.jpg)
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
![Page 36: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/36.jpg)
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
![Page 37: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/37.jpg)
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
![Page 38: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/38.jpg)
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
![Page 39: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/39.jpg)
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
![Page 40: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/40.jpg)
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
![Page 41: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/41.jpg)
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
![Page 42: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/42.jpg)
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
![Page 43: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/43.jpg)
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
![Page 44: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/44.jpg)
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
![Page 45: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/45.jpg)
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
![Page 46: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/46.jpg)
Generating Series for Partitions
Anna Weigandt Partition Identities and Quiver Representations
![Page 47: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/47.jpg)
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
![Page 48: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/48.jpg)
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
![Page 49: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/49.jpg)
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
![Page 50: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/50.jpg)
The Generating Series for Rectangles of Width k
∅ . . .
1 + qk + q2k + q3k + q4k + q5k + . . . =1
1− qk
Anna Weigandt Partition Identities and Quiver Representations
![Page 51: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/51.jpg)
The Generating Series for Rectangles of Width k
∅ . . .
1 + qk + q2k + q3k + q4k + q5k + . . . =1
1− qk
Anna Weigandt Partition Identities and Quiver Representations
![Page 52: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/52.jpg)
The Generating Series for Rectangles of Width k
∅ . . .
1 + qk + q2k + q3k + q4k + q5k + . . .
=1
1− qk
Anna Weigandt Partition Identities and Quiver Representations
![Page 53: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/53.jpg)
The Generating Series for Rectangles of Width k
∅ . . .
1 + qk + q2k + q3k + q4k + q5k + . . . =1
1− qk
Anna Weigandt Partition Identities and Quiver Representations
![Page 54: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/54.jpg)
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
![Page 55: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/55.jpg)
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
![Page 56: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/56.jpg)
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
![Page 57: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/57.jpg)
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
![Page 58: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/58.jpg)
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
![Page 59: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/59.jpg)
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
![Page 60: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/60.jpg)
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
![Page 61: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/61.jpg)
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
![Page 62: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/62.jpg)
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
![Page 63: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/63.jpg)
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
![Page 64: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/64.jpg)
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
![Page 65: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/65.jpg)
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
![Page 66: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/66.jpg)
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
![Page 67: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/67.jpg)
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
![Page 68: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/68.jpg)
Combinatorial Interpretation of the q-Binomial Coefficient
(x + y)11 = . . .+ yyxyxyyxxyy + . . .
Anna Weigandt Partition Identities and Quiver Representations
![Page 69: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/69.jpg)
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
![Page 70: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/70.jpg)
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
![Page 71: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/71.jpg)
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
![Page 72: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/72.jpg)
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
![Page 73: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/73.jpg)
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
![Page 74: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/74.jpg)
Durfee Squares and Rectangles
Anna Weigandt Partition Identities and Quiver Representations
![Page 75: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/75.jpg)
Durfee Squares
The Durfee square D(λ) is the largest j× j square partitioncontained in λ.
Anna Weigandt Partition Identities and Quiver Representations
![Page 76: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/76.jpg)
Durfee Squares
The Durfee square D(λ) is the largest j× j square partitioncontained in λ.
Anna Weigandt Partition Identities and Quiver Representations
![Page 77: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/77.jpg)
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
![Page 78: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/78.jpg)
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
![Page 79: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/79.jpg)
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
![Page 80: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/80.jpg)
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
![Page 81: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/81.jpg)
Proof of the Main Theorem
Anna Weigandt Partition Identities and Quiver Representations
![Page 82: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/82.jpg)
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
![Page 83: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/83.jpg)
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
![Page 84: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/84.jpg)
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
![Page 85: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/85.jpg)
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
![Page 86: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/86.jpg)
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
![Page 87: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/87.jpg)
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
![Page 88: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/88.jpg)
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
![Page 89: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/89.jpg)
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
![Page 90: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/90.jpg)
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
![Page 91: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/91.jpg)
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
![Page 92: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/92.jpg)
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
![Page 93: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/93.jpg)
λ(1)
s21
t11
Anna Weigandt Partition Identities and Quiver Representations
![Page 94: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/94.jpg)
λ(1)
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
![Page 95: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/95.jpg)
λ(2)
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
![Page 96: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/96.jpg)
λ(2)
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
![Page 97: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/97.jpg)
λ(2)
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
![Page 98: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/98.jpg)
λ(2)
t21
s21
Anna Weigandt Partition Identities and Quiver Representations
![Page 99: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/99.jpg)
λ(2)
t21t2
2
s21
Anna Weigandt Partition Identities and Quiver Representations
![Page 100: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/100.jpg)
λ(3)
t21t2
2
s31
Anna Weigandt Partition Identities and Quiver Representations
![Page 101: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/101.jpg)
λ(3)
t21t2
2
s31
Anna Weigandt Partition Identities and Quiver Representations
![Page 102: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/102.jpg)
λ(3)
t21t2
2
s31
Anna Weigandt Partition Identities and Quiver Representations
![Page 103: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/103.jpg)
λ(3)
t31
s31
Anna Weigandt Partition Identities and Quiver Representations
![Page 104: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/104.jpg)
λ(3)
t31t3
2t33
s31
s32
Anna Weigandt Partition Identities and Quiver Representations
![Page 105: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/105.jpg)
λ(4)
t31t3
2t33
Anna Weigandt Partition Identities and Quiver Representations
![Page 106: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/106.jpg)
λ(4)
t31t3
2t33
Anna Weigandt Partition Identities and Quiver Representations
![Page 107: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/107.jpg)
λ(4)
t31t3
2t33
Anna Weigandt Partition Identities and Quiver Representations
![Page 108: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/108.jpg)
λ(4)
t41
s41
Anna Weigandt Partition Identities and Quiver Representations
![Page 109: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/109.jpg)
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
![Page 110: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/110.jpg)
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
![Page 111: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/111.jpg)
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
![Page 112: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/112.jpg)
λ 7→ (µ,ν)
∅
Anna Weigandt Partition Identities and Quiver Representations
![Page 113: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/113.jpg)
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
![Page 114: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/114.jpg)
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
![Page 115: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/115.jpg)
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
![Page 116: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/116.jpg)
Merci!
Anna Weigandt Partition Identities and Quiver Representations
![Page 117: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/117.jpg)
Anna Weigandt Partition Identities and Quiver Representations
![Page 118: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/118.jpg)
What happens with a different choice of w?
Anna Weigandt Partition Identities and Quiver Representations
![Page 119: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/119.jpg)
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
![Page 120: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/120.jpg)
λ(1) w(1) = 1
s21
t11
Anna Weigandt Partition Identities and Quiver Representations
![Page 121: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/121.jpg)
λ(1) w(1) = 1
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
![Page 122: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/122.jpg)
λ(2) w(2) = 12
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
![Page 123: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/123.jpg)
λ(2) w(2) = 12
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
![Page 124: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/124.jpg)
λ(2) w(2) = 12
t11
s21
Anna Weigandt Partition Identities and Quiver Representations
![Page 125: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/125.jpg)
λ(2) w(2) = 12
t21
s21
Anna Weigandt Partition Identities and Quiver Representations
![Page 126: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/126.jpg)
λ(2) w(2) = 12
t21t2
2
s21
Anna Weigandt Partition Identities and Quiver Representations
![Page 127: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/127.jpg)
λ(3) w(3) = 213
t21 t2
2
s31
Anna Weigandt Partition Identities and Quiver Representations
![Page 128: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/128.jpg)
λ(3) w(3) = 213
t21 t2
2
s31
Anna Weigandt Partition Identities and Quiver Representations
![Page 129: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/129.jpg)
λ(3) w(3) = 213
t21 t2
2
s31
Anna Weigandt Partition Identities and Quiver Representations
![Page 130: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/130.jpg)
λ(3) w(3) = 213
t31
s31
Anna Weigandt Partition Identities and Quiver Representations
![Page 131: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/131.jpg)
λ(3) w(3) = 213
t32t3
1t33
s32
s31
Anna Weigandt Partition Identities and Quiver Representations
![Page 132: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/132.jpg)
λ(4) w(4) = 3124
t31t3
2 t33
Anna Weigandt Partition Identities and Quiver Representations
![Page 133: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/133.jpg)
λ(4) w(4) = 3124
t31t3
2 t33
Anna Weigandt Partition Identities and Quiver Representations
![Page 134: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/134.jpg)
λ(4) w(4) = 3124
t31t3
2 t33
Anna Weigandt Partition Identities and Quiver Representations
![Page 135: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/135.jpg)
λ(4) w(4) = 3124
t41
s41
Anna Weigandt Partition Identities and Quiver Representations
![Page 136: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/136.jpg)
λ(4) w(4) = 3124
t41t4
2 t43t4
4
s41
s42
s43
Anna Weigandt Partition Identities and Quiver Representations
![Page 137: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/137.jpg)
The Associated Lacing Diagram
versus
Anna Weigandt Partition Identities and Quiver Representations
![Page 138: Partition Identities and Quiver Representationsweigandt/QuiverPartSherbrooke.pdf · 2020. 5. 19. · Partition Identities and Quiver Representations Anna Weigandt University of Illinois](https://reader036.vdocuments.us/reader036/viewer/2022071411/61076742cd845a0f096fd011/html5/thumbnails/138.jpg)
The Associated Lacing Diagram
versus
Anna Weigandt Partition Identities and Quiver Representations