Partition Identities and Quiver Representations Anna Weigandt University of Illinois at Urbana-Champaign weigndt2@illinois.edu November 20th, 2017 Based on joint work with Rich´ ard Rim´ anyi and Alexander Yong arXiv:1608.02030 Anna Weigandt Partition Identities and Quiver Representations
Overview This project provides an elementary explanation for a quantum dilogarithm identity due to M. Reineke. We use generating function techniques to establish a related identity, which is a generalization of the Euler-Gauss identity. This reduces to an equivalent form of Reineke’s identity in type A. Anna Weigandt Partition Identities and Quiver Representations
Representations of Quivers Anna Weigandt Partition Identities and Quiver Representations
Quivers A quiver Q = ( Q 0 , Q 1 ) is a directed graph with vertices : i ∈ Q 0 edges : a : i → j ∈ Q 1 Anna Weigandt Partition Identities and Quiver Representations
The Definition A representation of Q is an assignment of a: vector space V i to each vertex i ∈ Q 0 and a linear transformation f a : V i → V j to each arrow i − → j ∈ Q 1 dim ( V ) = ( dim V i ) i ∈ Q 0 is the dimension vector of V . Anna Weigandt Partition Identities and Quiver Representations
The Representation Space Fix d ∈ N Q 0 . The representation space is ⊕ Rep Q ( d ) := Mat( d ( i ) , d ( j )) . a − → j ∈ Q 1 i Let ∏ GL Q ( d ) := GL( d ( i )) . i ∈ Q 0 GL Q ( d ) acts on Rep Q ( d ) by base change at each vertex. Orbits of this action are in bijection with isomorphism classes of d dimensional representations. Anna Weigandt Partition Identities and Quiver Representations
Dynkin Quivers A quiver is Dynkin if its underlying graph is of type ADE : Anna Weigandt Partition Identities and Quiver Representations
Gabriel’s Theorem Theorem ([Gab75]) Dynkin quivers have finitely many isomorphism classes of indecomposable representations. For type A , indecomposables V [ i , j ] are indexed by intervals . Anna Weigandt Partition Identities and Quiver Representations
Lacing Diagrams A lacing diagram ([ADF85]) L is a graph so that: the vertices are arranged in n columns labeled 1 , 2 , . . . , n the edges between adjacent columns form a partial matching. Anna Weigandt Partition Identities and Quiver Representations
Lacing Diagrams A lacing diagram ([ADF85]) L is a graph so that: the vertices are arranged in n columns labeled 1 , 2 , . . . , n the edges between adjacent columns form a partial matching. Idea: Lacing diagrams are a way to visually encode representations of an A n quiver. Anna Weigandt Partition Identities and Quiver Representations
The Role of Lacing Diagrams in Representation Theory When Q is a type A quiver, a lacing diagram can be interpreted as a sequence of partial permutation matrices which form a representation V L of Q . 4 3 2 1 0 0 0 [ 1 1 0 ] 0 0 0 0 0 0 , , 0 0 0 1 0 0 1 0 0 0 0 0 1 0 See [KMS06] for the equiorientated case and [BR04] for arbitrary orientations . Anna Weigandt Partition Identities and Quiver Representations
Equivalence Classes of Lacing Diagrams Two lacing diagrams are equivalent if one can be obtained from the other by permuting vertices within a column. Anna Weigandt Partition Identities and Quiver Representations
Equivalence Classes of Lacing Diagrams Two lacing diagrams are equivalent if one can be obtained from the 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
Strands A strand is a connected component of L . Anna Weigandt Partition Identities and Quiver Representations
Strands A strand is a connected component of L . m [ i , j ] ( L ) = |{ strands starting at column i and ending at column j }| Example: m [1 , 2] ( L ) = 2 Anna Weigandt Partition Identities and Quiver Representations
Strands A strand is a connected component of L . m [ i , j ] ( L ) = |{ strands starting at column i and ending at column j }| m [1 , 2] ( L ) = 2 Example: Anna Weigandt Partition Identities and Quiver Representations
Strands A strand is a connected component of L . m [ i , j ] ( L ) = |{ strands starting at column i and ending at column j }| m [4 , 4] ( L ) = 1 Example: Anna Weigandt Partition Identities and Quiver Representations
Strands Strands record the decomposition of V L into indecomposable representations: ⊕ m [ i , j ] ( L ) V L ∼ = ⊕ V [ i , j ] Example: V L ∼ = V ⊕ 2 [1 , 2] ⊕ V [2 , 3] ⊕ V [2 , 4] ⊕ V [3 , 3] ⊕ V [4 , 4] Anna Weigandt Partition Identities and Quiver Representations
Reineke’s Identities Anna Weigandt Partition Identities and Quiver Representations
The Quantum Dilogarithm Series ∞ q k 2 / 2 z k ∑ E ( z ) = (1 − q )(1 − q 2 ) . . . (1 − q k ) k =0 Anna Weigandt Partition Identities and Quiver Representations
The Quantum Algebra of a Quiver The Quantum Algebra A Q is an algebra over Q ( q 1 / 2 ) with generators: { y d : d ∈ N Q 0 } multiplication: 1 2 ( χ ( d 2 , d 1 ) − χ ( d 1 , d 2 )) y d 1 + d 2 y d 1 y d 2 = q The Euler form χ : N Q 0 × N Q 0 → Z ∑ ∑ χ ( d 1 , d 2 ) = d 1 ( i ) d 2 ( i ) − d 1 ( i ) d 2 ( j ) a i ∈ Q 0 − → j ∈ Q 1 i Anna Weigandt Partition Identities and Quiver Representations
Reineke’s Identity Given a representation V , we’ll write d V as a shorthand for d dim ( V ) . For a Dynkin quiver, it is possible to fix a choice of ordering on the simple representations: α 1 , . . . , α n indecomposable representations: β 1 , . . . , β N so that E ( y d α 1 ) · · · E ( y d α n ) = E ( y d β 1 ) · · · E ( y d β N ) (1) (Original proof given by [Rei10], see [Kel11] for exposition and a sketch of the proof.) Anna Weigandt Partition Identities and Quiver Representations
Reformulation Looking at the coefficient of y d on each side, this is equivalent to the following infinite family of identities ([Rim13]): n N 1 1 ∏ ∑ q codim C ( η ) ∏ = . ( q ) d ( i ) ( q ) m β i ( η ) η i =1 i =1 where the sum is over orbits η in Rep Q ( d ) and m β ( η ) is the multiplicity of β in V ∈ η . Here, ( q ) k = (1 − q ) · · · (1 − q k ) is the q -shifted factorial. Anna Weigandt Partition Identities and Quiver Representations
Some Bookkeeping Fix a sequence of permutations w = ( w (1) , . . . , w ( n ) ), so that w ( i ) ∈ S i and w ( i ) ( i ) = i . Let s j i ( L ) = m [ i , j − 1] ( L ) and t k i ( L ) = m [ i , k ] ( L ) + m [ i , k +1] ( L ) + . . . + m [ i , n ] ( L ) . Define the Durfee statistic : ∑ s k w ( k ) ( i ) ( L ) t k r w ( L ) = w ( k ) ( j ) ( L ) . 1 ≤ i < j ≤ k ≤ n Anna Weigandt Partition Identities and Quiver Representations
Some Bookkeeping Fix a sequence of permutations w = ( w (1) , . . . , w ( n ) ), so that w ( i ) ∈ S i and w ( i ) ( i ) = i . Let s j i ( L ) = m [ i , j − 1] ( L ) and t k i ( L ) = m [ i , k ] ( L ) + m [ i , k +1] ( L ) + . . . + m [ i , n ] ( L ) . Define the Durfee statistic : ∑ s k w ( k ) ( i ) ( L ) t k r w ( L ) = w ( k ) ( j ) ( L ) . 1 ≤ i < j ≤ k ≤ n The above statistics are all constant on equivalence classes of lacing diagrams. Anna Weigandt Partition Identities and Quiver Representations
Theorem (Rim´ anyi, Weigandt, Yong, 2016) Fix a dimension vector d = ( d (1) , . . . , d ( n )) and let w be as before. Then n n k − 1 [ t k i ( η ) + s k i ( η ) ] 1 1 ∏ ∑ ∏ ∏ q r w ( η ) = s k ( q ) d ( k ) ( q ) t k i ( η ) k ( η ) q k =1 η ∈ L ( d ) k =1 i =1 [ j + k ] q is the q -binomial coefficient and ( q ) k the q - shifted k factorial . 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 ∑ q wt ( s ) . G ( S , q ) = s ∈ S Anna Weigandt Partition Identities and Quiver Representations
Partitions An integer partition is an ordered list of decreasing integers: λ = λ 1 ≥ λ 2 ≥ . . . ≥ λ ℓ ( λ ) > 0 We will typically represent a partition by its Young diagram : We weight a partition by counting the boxes in its Young diagram. Anna Weigandt Partition Identities and Quiver Representations
Generating Series for Partitions Anna Weigandt Partition Identities and Quiver Representations
Generating Series for Partitions Anna Weigandt Partition Identities and Quiver Representations
Generating Series for Partitions ∞ 1 1 ∏ = (1 − q k ) ( q ) ∞ k =1 Anna Weigandt Partition Identities and Quiver Representations
Generating Series for Partitions ∞ ∞ 1 1 (1 + q k + q 2 k + q 3 k + . . . ) ∏ ∏ = (1 − q k ) = ( q ) ∞ k =1 k =1 Anna Weigandt Partition Identities and Quiver Representations
Generating Series for Partitions ∞ ∞ 1 1 (1 + q k + q 2 k + q 3 k + . . . ) ∏ ∏ = (1 − q k ) = ( q ) ∞ k =1 k =1 q 16 = q 1 · 1 · q 3 · 3 · 1 · 1 · q 6 · 1 · 1 · . . . Anna Weigandt Partition Identities and Quiver Representations
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