From Clusters to Quivers From Cluster Algebras to Quiver Grassmannians Dylan Rupel Michigan State University April 26, 2019 Maurice Auslander Distinguished Lectures and International Conference Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 1 / 21
From Clusters to Quivers Main Result Cell Decompositions for Rank Two Quiver Grassmannians MetaTheorem/Conjecture The combinatorics of compatible subsets of maximal Dyck paths controls the geometry of quiver Grassmannians. Theorem (R.-Weist) For k ∈ Z \ { 1 , 2 } and e = ( e 1 , e 2 ) ∈ Z 2 ≥ 0 , the quiver Grassmannian Gr e ( M k ) admits a cell decomposition (affine paving) whose affine cells are naturally labeled by compatible subsets S ∈ C k with � u k − 1 − e 1 if k ≥ 3 | S ∩ V k | = e 2 | S ∩ H k | = u 1 − k − e 1 if k ≤ 0 Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 2 / 21
From Clusters to Quivers Cluster Algebras A Simple Example - Rank Two Fix an integer n ≥ 2 . Define cluster variables x k ∈ Q ( x 1 , x 2 ) , k ∈ Z , recursively by x k − 1 x k +1 = x n k + 1 . The first few cluster variables are computed as follows: x 3 = x n 2 + 1 x 1 2 + 1) n + x n x 4 = x n = ( x n 3 + 1 1 x n x 2 1 x 2 x 5 = x n 4 + 1 = N ( x 1 , x 2 ) x n 2 − 1 x 3 x n 1 2 Here N ( x 1 , x 2 ) ∈ Z [ x 1 , x 2 ] and so a non-trivial cancellation has occurred. Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 3 / 21
From Clusters to Quivers Cluster Algebras Laurent Phenomenon Fix an integer n ≥ 2 . Define cluster variables x k ∈ Q ( x 1 , x 2 ) , k ∈ Z , recursively by x k − 1 x k +1 = x n k + 1 . Theorem (Fomin-Zelevinsky, Laurent Phenomenon) Each cluster variable x k ∈ Q ( x 1 , x 2 ) , k ∈ Z , can be written as x k = N k ( x 1 , x 2 ) , x d k, 1 x d k, 2 1 2 for some polynomial N k ( x 1 , x 2 ) ∈ Z [ x 1 , x 2 ] with nonzero constant term and some denominator vector d k = ( d k, 1 , d k, 2 ) ∈ Z 2 . First Goal: Understand these Laurent expansions of the cluster variables Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 3 / 21
From Clusters to Quivers Cluster Algebras Denominator Vectors The denominators are relatively easy to describe: define Chebyshev polynomials u m = u m ( n ) ∈ Z for m ∈ Z recursively by u 1 = 0 , u 2 = 1 , u m +1 = nu m − u m − 1 . Proposition For k ∈ Z , the denominator vector of x k is given by � ( u k − 1 , u k − 2 ) if k ≥ 2 d k = ( u 1 − k , u 2 − k ) if k ≤ 1 Goal: Understand the numerators N k ( x 1 , x 2 ) of the cluster variables x k . I will present two approaches: one geometric and one combinatorial (explaining the relationship between them is the ultimate goal of this talk) Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 4 / 21
From Clusters to Quivers Geometric Construction - Quiver Representations Basic Definitions n Let Q n = 1 ← − 2 be the n -Kronecker quiver with vertex set { 1 , 2 } and arrows α j , j = 1 , . . . , n , from vertex 2 to vertex 1 . A representation M = ( M 1 , M 2 , M α j ) of Q n consists of the following: C -vector spaces M i for i = 1 , 2 C -linear maps M α j : M 2 → M 1 for j = 1 , . . . , n Write dim ( M ) = ( dim M 1 , dim M 2 ) for the dimension vector of M Given representations M = ( M 1 , M 2 , M α j ) and N = ( N 1 , N 2 , N α j ) , a morphism θ : M → N consists of linear maps θ i : M i → N i such that θ 1 ◦ M α j = N α j ◦ θ 2 for j = 1 , . . . , n Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 5 / 21
From Clusters to Quivers Geometric Construction - Quiver Representations Quiver Grassmannians A subrepresentation E ⊂ M consists of subspaces E i ⊂ M i such that M α j ( E 2 ) ⊂ E 1 for j = 1 , . . . , n Definition Given a dimension vector e = ( e 1 , e 2 ) ∈ Z 2 ≥ 0 , the quiver Grassmannian Gr e ( M ) is the set of all subrepresentations E ⊂ M with dim ( E ) = e . Lemma Gr e ( M ) is a projective variety Proof. Gr e ( M ) is naturally identified with a subset of the product of ordinary vector space Grassmannians Gr e 1 ( M 1 ) × Gr e 2 ( M 2 ) which is projective. The requirements M α j ( E 2 ) ⊂ E 1 give closed conditions cutting out the quiver Grassmannian Gr e ( M ) . Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 6 / 21
From Clusters to Quivers Geometric Construction - Quiver Representations Quiver Grassmannians Theorem (Reineke, Huisgen-Zimmermann, Hille, Ringel) Every projective variety is isomorphic to a quiver Grassmannian Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 7 / 21
From Clusters to Quivers Geometric Construction - Quiver Representations Quiver Grassmannians Theorem (Reineke, Huisgen-Zimmermann, Hille, Ringel) Every projective variety is isomorphic to a quiver Grassmannian of Q n for any n ≥ 3 . Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 7 / 21
From Clusters to Quivers Geometric Construction - Quiver Representations Quiver Grassmannians Theorem (Reineke, Huisgen-Zimmermann, Hille, Ringel) Every projective variety is isomorphic to a quiver Grassmannian of Q n for any n ≥ 3 . Moral: one cannot expect great control over the geometry of quiver Grassmannians without imposing conditions on M or e . A representation M is rigid if Ext 1 ( M, M ) = 0 . In this case, representations isomorphic to M form a dense subset of the moduli space of representations with dimension vector dim ( M ) . Theorem (Caldero-Reineke) Assume Gr e ( M ) is nonempty and M is rigid. Then Gr e ( M ) is a smooth projective variety. Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 7 / 21
From Clusters to Quivers Geometric Construction - Quiver Representations Geometric Construction of Rank Two Cluster Variables Theorem (Bernstein-Gelfand-Ponamarev, Dlab-Ringel) For k ∈ Z \ { 1 , 2 } , there exists a unique (up to isomorphism) indecomposable rigid representation M k of Q n with dimension vector � ( u k − 1 , u k − 2 ) if k ≥ 2 d k = ( u 1 − k , u 2 − k ) if k ≤ 1 Theorem (Caldero-Chapoton, Caldero-Keller, R./Qin (quantum case)) Each cluster variable x k ∈ Q ( x 1 , x 2 ) for k ∈ Z \ { 1 , 2 } is a generating function for the Euler characteristics of the quiver Grassmannians for M k : x − u k − 1 x − u k − 2 x n ( u k − 1 − e 1 ) x ne 2 � � � χ Gr e ( M k ) if k ≥ 3 1 2 1 2 e ∈ Z 2 ≥ 0 x k = x − u 1 − k x − u 2 − k x n ( u 1 − k − e 1 ) x ne 2 � � � χ Gr e ( M k ) if k ≤ 0 1 2 1 2 e ∈ Z 2 ≥ 0 Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 8 / 21
From Clusters to Quivers Combinatorial Construction - Compatible Subsets Maximal Dyck Paths Recall the denominator/dimension vectors for k ∈ Z : � ( u k − 1 , u k − 2 ) if k ≥ 2 d k = ( u 1 − k , u 2 − k ) if k ≤ 1 Definition For k ∈ Z \ { 1 , 2 } , write D k for the maximal Dyck path in the lattice rectangle in Z 2 with corner vertices (0 , 0) and d k . D k is a lattice path beginning at (0 , 0) , taking East and North steps to end at d k , and never passing above the main diagonal. Any lattice point above D k also lies above the main diagonal. Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 9 / 21
From Clusters to Quivers Combinatorial Construction - Compatible Subsets Maximal Dyck Paths Recall the denominator/dimension vectors for k ∈ Z : � ( u k − 1 , u k − 2 ) if k ≥ 2 d k = ( u 1 − k , u 2 − k ) if k ≤ 1 Definition For k ∈ Z \ { 1 , 2 } , write D k for the maximal Dyck path in the lattice rectangle in Z 2 with corner vertices (0 , 0) and d k . Write H k and V k for the sets of horizontal and vertical edges of D k . The edges H k ⊔ V k are naturally ordered along the Dyck path D k from (0 , 0) to d k . Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 9 / 21
From Clusters to Quivers Combinatorial Construction - Compatible Subsets Combinatorics of Maximal Dyck Paths For n = 3 , we have d 3 = (1 , 0) , d 4 = (3 , 1) , d 5 = (8 , 3) , d 6 = (21 , 8) The associated maximal Dyck paths are shown below: D 3 = D 4 = D 5 = D 6 = Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 10 / 21
From Clusters to Quivers Combinatorial Construction - Compatible Subsets Combinatorics of Maximal Dyck Paths For n = 3 , we have d 3 = (1 , 0) , d 4 = (3 , 1) , d 5 = (8 , 3) , d 6 = (21 , 8) The associated maximal Dyck paths are shown below: D 3 = D 4 = D 5 = D 6 = Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 10 / 21
From Clusters to Quivers Combinatorial Construction - Compatible Subsets Combinatorics of Maximal Dyck Paths For n = 3 , we have d 3 = (1 , 0) , d 4 = (3 , 1) , d 5 = (8 , 3) , d 6 = (21 , 8) The associated maximal Dyck paths are shown below: D 3 = D 4 = D 5 = D 6 = Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 10 / 21
From Clusters to Quivers Combinatorial Construction - Compatible Subsets Combinatorics of Maximal Dyck Paths For n = 3 , we have d 3 = (1 , 0) , d 4 = (3 , 1) , d 5 = (8 , 3) , d 6 = (21 , 8) The associated maximal Dyck paths are shown below: D 3 = D 4 = D 5 = D 6 = Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 10 / 21
From Clusters to Quivers Combinatorial Construction - Compatible Subsets Combinatorics of Maximal Dyck Paths D 4 = D 5 = D 6 = Proposition For k ≥ 5 , the maximal Dyck path D k can be constructed by concatenating n − 1 copies of D k − 1 followed by a copy of D k − 1 with its first D k − 2 removed. Dylan Rupel (MSU) From Clusters to Quivers April 26, 2019 10 / 21
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