Partition Identities and Quiver Representations Anna Weigandt University of Illinois at Urbana-Champaign [email protected]January 23rd, 2017 Based on joint work with Rich´ ard Rim´ anyi and Alexander Yong arXiv:1608.02030 Anna Weigandt Partition Identities and Quiver Representations
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Anna Weigandt Partition Identities and Quiver Representations
Statistics on Lacing Diagrams
tki (L) = |{strands starting at column i which use column k}|
Example: t32 (L) = 2
Anna Weigandt Partition Identities and Quiver Representations
Statistics on Lacing Diagrams
tki (L) = |{strands starting at column i which use column k}|
Example: t32 (L) = 2
Anna Weigandt Partition Identities and Quiver Representations
Statistics on Lacing Diagrams
tki (L) = |{strands starting at column i which use column k}|
Example: t42 (L) = 1
Anna Weigandt Partition Identities and Quiver Representations
Statistics on Lacing Diagrams
ski (L) = m[i ,k−1](L)
Example: s42 (L) = 1
Anna Weigandt Partition Identities and Quiver Representations
Statistics on Lacing Diagrams
ski (L) = m[i ,k−1](L)
Example: s42 (L) = 1
Anna Weigandt Partition Identities and Quiver Representations
The Durfee Statistic
Let L(d) = {[L] : dim(L) = d}.
Fix a sequence of permutations w = (w (1), . . . ,w (n)), so thatw (i) ∈ Si and w (i)(i) = i .
Define the Durfee statistic:
rw(L) =∑
1≤i<j≤k≤n
skw (k)(i)
(L)tkw (k)(j)
(L).
Anna Weigandt Partition Identities and Quiver Representations
The Durfee Statistic
Let L(d) = {[L] : dim(L) = d}.
Fix a sequence of permutations w = (w (1), . . . ,w (n)), so thatw (i) ∈ Si and w (i)(i) = i .
Define the Durfee statistic:
rw(L) =∑
1≤i<j≤k≤n
skw (k)(i)
(L)tkw (k)(j)
(L).
Anna Weigandt Partition Identities and Quiver Representations
Theorem (Rimanyi, 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)tkk (η)
k−1∏i=1
[tki (η) + ski (η)
ski (η)
]q
Note:[j+k
k
]qis the q-binomial coefficient.
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 that|{s ∈ S : wt(s) = k}| <∞
for each k ∈ N.
The generating series for S is
G (S , q) =∑s∈S
qwt(s).
Anna Weigandt Partition Identities and Quiver Representations
Partitions
An integer partition is an ordered list of decreasing integers:
λ = λ1 ≥ λ2 ≥ . . . ≥ λℓ(λ) ≥ 0
We will typically represent a partition by its Young diagram:
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
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
)x iy j .
q-Binomial Coefficient:If x and y are q commuting (yx = qxy),
(x + y)n =∑i+j=n
[i + j
i
]q
x iy j .
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
)x iy j .
q-Binomial Coefficient:If x and y are q commuting (yx = qxy),
(x + y)n =∑i+j=n
[i + j
i
]q
x iy j .
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 + . . .
x
y y x
y x
y 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 + . . .
x
y y x
y x
y 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 + . . .
x
y y x
y x
y 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 + . . .
x
y y x
y x
y 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 + . . .
x
y y x
y x
y 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≥0
R(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≥0
R(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 (Rimanyi, 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)tkk (η)
k−1∏i=1
[tki (η) + ski (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 (Rimanyi, 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)tkk (η)
k−1∏i=1
[tki (η) + ski (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=1
G (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)
T (η)
T (η) = R(η)× P(η)
G (T , q) =∑
η∈L(d)
G (T (η), 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)
T (η)
T (η) = R(η)× P(η)
G (T , q) =∑
η∈L(d)
G (T (η), 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(skw (k)(i)
(η), tkw (k)(j)
(η)), 1 ≤ i < j ≤ k ≤ n}
Recall the Durfee Statistic:
rw(η) =∑
1≤i<j≤k≤n
skw (k)(i)
(η)tkw (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(skw (k)(i)
(η), tkw (k)(j)
(η)), 1 ≤ i < j ≤ k ≤ n}
Recall the Durfee Statistic:
rw(η) =∑
1≤i<j≤k≤n
skw (k)(i)
(η)tkw (k)(j)
(η)
G (R(η), q) = qrw(η)
Anna Weigandt Partition Identities and Quiver Representations
Partitions
For convenience, write skk (η) =∞.
P(η) = {ν = (νki ) : νki ∈ P(skw (k)(i)
(η), tkw (k)(i)
(η)), 1 ≤ i ≤ k ≤ n}
G (P(η), q) =n∏
k=1
1
(q)tkk (η)
k−1∏i=1
[tki (η) + ski (η)
ski (η)
]q
Anna Weigandt Partition Identities and Quiver Representations
Partitions
For convenience, write skk (η) =∞.
P(η) = {ν = (νki ) : νki ∈ P(skw (k)(i)
(η), tkw (k)(i)
(η)), 1 ≤ i ≤ k ≤ n}
G (P(η), q) =n∏
k=1
1
(q)tkk (η)
k−1∏i=1
[tki (η) + ski (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)
t21t22
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(3)
t21t22
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3)
t21t22
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3)
t21t22
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3)
t31
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3)
t31t32t33
s31
s32
Anna Weigandt Partition Identities and Quiver Representations
λ(4)
t31t32t33
Anna Weigandt Partition Identities and Quiver Representations
λ(4)
t31t32t33
Anna Weigandt Partition Identities and Quiver Representations
λ(4)
t31t32t33
Anna Weigandt Partition Identities and Quiver Representations
λ(4)
t41
s41
Anna Weigandt Partition Identities and Quiver Representations
λ(4)
t41t42t43t44
s41s42
s43
Anna Weigandt Partition Identities and Quiver Representations
These Parameters are Well Defined
Lemma
There exists a unique η ∈ L(d) so that ski (η) = ski and tkj (η) = tkjfor all i , j , k.
Anna Weigandt Partition Identities and Quiver Representations
A Recursion
For any η,ski (η) + tki (η) = tk−1
i (η) (4)
The parameters defined by Durfee rectangles satisfy the sameequations:
t11 t21t22
s21
t31t32t33
s31
s32
t41t42t43t44s41s42s43
λ 7→ (µ,ν) ∈ T (η) ⊆ T
Anna Weigandt Partition Identities and Quiver Representations
A Recursion
For any η,ski (η) + tki (η) = tk−1
i (η) (4)
The parameters defined by Durfee rectangles satisfy the sameequations:
t11 t21t22
s21
t31t32t33
s31
s32
t41t42t43t44s41s42s43
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
λ 7→ (µ,ν)
∅
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
t21t22
s21
Anna Weigandt Partition Identities and Quiver Representations
λ(3) w (3) = 213
t21 t22
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3) w (3) = 213
t21 t22
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(3) w (3) = 213
t21 t22
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
t32t31t33
s32
s31
Anna Weigandt Partition Identities and Quiver Representations
λ(4) w (4) = 3124
t31t32 t33
Anna Weigandt Partition Identities and Quiver Representations
λ(4) w (4) = 3124
t31t32 t33
Anna Weigandt Partition Identities and Quiver Representations
λ(4) w (4) = 3124
t31t32 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
t41t42 t43t44
s41s42
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
Connection with Reineke’s Identity(in type A)
Anna Weigandt Partition Identities and Quiver Representations
Simplifying the Identity
Lets think about the blue terms:
1
(q)t11
[s21 + t21
s21
]q
[s31 + t31
s31
]q
[s41 + t41
s41
]q
=
(1
(q)t11
)((q)s21+t21
(q)s21(q)t21
)((q)s31+t31
(q)s31(q)t31
)((q)s41+t41
(q)s41(q)t41
)
=1
(q)s21(q)s31
(q)s41(q)t41
=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
Lets think about the blue terms:
1
(q)t11
[s21 + t21
s21
]q
[s31 + t31
s31
]q
[s41 + t41
s41
]q
=
(1
(q)t11
)((q)s21+t21
(q)s21(q)t21
)((q)s31+t31
(q)s31(q)t31
)((q)s41+t41
(q)s41(q)t41
)
=1
(q)s21(q)s31
(q)s41(q)t41
=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
Lets think about the blue terms:
1
(q)t11
[s21 + t21
s21
]q
[s31 + t31
s31
]q
[s41 + t41
s41
]q
=
(1
(q)t11
)((q)s21+t21
(q)s21(q)t21
)((q)s31+t31
(q)s31(q)t31
)((q)s41+t41
(q)s41(q)t41
)
=1
(q)s21(q)s31
(q)s41(q)t41
=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
Lets think about the blue terms:
1
(q)t11
[s21 + t21
s21
]q
[s31 + t31
s31
]q
[s41 + t41
s41
]q
=
(1
(q)t11
)((q)s21+t21
(q)s21(q)t21
)((q)s31+t31
(q)s31(q)t31
)((q)s41+t41
(q)s41(q)t41
)
=1
(q)s21(q)s31
(q)s41(q)t41
=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
Lets think about the blue terms:
1
(q)t11
[s21 + t21
s21
]q
[s31 + t31
s31
]q
[s41 + t41
s41
]q
=
(1
(q)t11
)((q)s21+t21
(q)s21(q)t21
)((q)s31+t31
(q)s31(q)t31
)((q)s41+t41
(q)s41(q)t41
)
=1
(q)s21(q)s31
(q)s41(q)t41
=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 (Rimanyi, 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 (Rimanyi, 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 Permutations
We 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 (Rimanyi, 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
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. Rimanyi.
A formula for non-equioriented quiver orbits of type a.
arXiv preprint math/0412073, 2004.
M. Brion.
Representations of quivers.
2008.
P. Gabriel.
Finite representation type is open.
In Representations of algebras, pages 132–155. Springer, 1975.
Anna Weigandt Partition Identities and Quiver Representations
References II
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., Zurich, 2011.
A. Knutson, E. Miller and M. Shimozono.
Four positive formulae for type A quiver polynomials.
Inventiones mathematicae, 166(2):229–325, 2006.
M. Reineke.
Poisson automorphisms and quiver moduli.
Journal of the Institute of Mathematics of Jussieu, 9(03):653–667, 2010.
Anna Weigandt Partition Identities and Quiver Representations
References III
R. Rimanyi.
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