From dimers in the disc to cluster categories arXiv:1912.12475 and - PowerPoint PPT Presentation

From dimers in the disc to cluster categories arXiv:1912.12475 and work in progress with İ. Çanakçı and A. King Matthew Pressland University of Leeds Dimers in Combinatorics and Cluster Algebras University of Michigan

Dimer models 1 7 We begin with a dimer model D 2 (a bipartite plabic graph) in the disc. The dimer model should be consistent, 6 meaning that this strands obtained by following the rules of the road 3 form a Postnikov diagram. The only non-automatic condition here is 5 4 that strands which cross twice should be oppositely oriented between these crossings—this also rules out closed strands in the interior. The dimer model has a chirality k “ p k ‚ , k ˝ q with k ‚ ` k ˝ “ n , the number of boundary marked points, and a permutation σ D of these points. This data determines a number of further geometric and algebraic objects, which we will explore.

Dimer models 1 7 We begin with a dimer model D 2 (a bipartite plabic graph) in the disc. The dimer model should be consistent, 6 meaning that this strands obtained by following the rules of the road 3 form a Postnikov diagram. The only non-automatic condition here is 5 4 that strands which cross twice should be oppositely oriented between these crossings—this also rules out closed strands in the interior. The dimer model has a chirality k “ p k ‚ , k ˝ q with k ‚ ` k ˝ “ n , the number of boundary marked points, and a permutation σ D of these points. This data determines a number of further geometric and algebraic objects, which we will explore.

The quiver 1 7 The dimer model D cuts the disk 2 into regions, and thus determines a quiver Q D with 6 ( Q 0 ) vertices corresponding to the regions 3 ( Q 1 ) arrows corresponding to edges, oriented with the black vertex on 5 4 the left. The vertices and arrows on the boundary—marked in blue and called frozen —sometimes play a different role to the others. In the Postnikov diagram, the vertices correspond to alternating regions, and the arrows to crossings, with their natural orientation.

The quiver 1 ˛ ˛ ˛ ˛ ˛ ˛ ˛ 7 ˛ ˛ ˛ ˛ ˛ ˛ ˛ The dimer model D cuts the disk 2 into regions, and thus determines ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ a quiver Q D with 6 ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ ( Q 0 ) vertices corresponding to the ˛ regions 3 ˛ ˛ ˛ ˛ ˛ ˛ ˛ ( Q 1 ) arrows corresponding to edges, ˛ ˛ ˛ ˛ ˛ ˛ ˛ oriented with the black vertex on 5 4 the left. ˛ ˛ ˛ ˛ ˛ ˛ ˛ The vertices and arrows on the boundary—marked in blue and called frozen —sometimes play a different role to the others. In the Postnikov diagram, the vertices correspond to alternating regions, and the arrows to crossings, with their natural orientation.

The quiver 1 ˛ ˛ ˛ ˛ ˛ ˛ ˛ 7 ˛ ˛ ˛ ˛ ˛ ˛ ˛ The dimer model D cuts the disk 2 into regions, and thus determines ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ a quiver Q D with 6 ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ ( Q 0 ) vertices corresponding to the ˛ regions 3 ˛ ˛ ˛ ˛ ˛ ˛ ˛ ( Q 1 ) arrows corresponding to edges, ˛ ˛ ˛ ˛ ˛ ˛ ˛ oriented with the black vertex on 5 4 the left. ˛ ˛ ˛ ˛ ˛ ˛ ˛ The vertices and arrows on the boundary—marked in blue and called frozen —sometimes play a different role to the others. In the Postnikov diagram, the vertices correspond to alternating regions, and the arrows to crossings, with their natural orientation.

The quiver 1 ˛ ˛ ˛ ˛ ˛ ˛ ˛ 7 ˛ ˛ ˛ ˛ ˛ ˛ ˛ The dimer model D cuts the disk 2 into regions, and thus determines ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ a quiver Q D with 6 ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ ( Q 0 ) vertices corresponding to the ˛ regions 3 ˛ ˛ ˛ ˛ ˛ ˛ ˛ ( Q 1 ) arrows corresponding to edges, ˛ ˛ ˛ ˛ ˛ ˛ ˛ oriented with the black vertex on 5 4 the left. ˛ ˛ ˛ ˛ ˛ ˛ ˛ The vertices and arrows on the boundary—marked in blue and called frozen —sometimes play a different role to the others. In the Postnikov diagram, the vertices correspond to alternating regions, and the arrows to crossings, with their natural orientation.

A cluster algebra The permutation σ D is a Grassmann permutation, and hence determines a particular open positroid subvariety Π p σ D q Ď Gr n k ‚ of the Grassmannian of k ‚ -dimensional subspaces of C n [Postnikov]. It also determines a cluster algebra A D , with invertible frozen variables, via the quiver Q D . Theorem (Serhiyenko–Sherman-Bennett–Williams, Galashin–Lam) „ There is an isomorphism A D Ñ C r Π p σ D qs , mapping the initial cluster variables to restrictions of Plücker coordinates. For σ D : i ÞÑ i ` k ˝ mod n (the uniform permutation ), the variety Π p σ D q is dense in Gr n k ‚ , and the cluster algebra with non-invertible frozen variables n attached to Q D is isomorphic to the homogeneous coordinate ring C r x Gr k ‚ s . [Scott] In this case, Jensen–King–Su have categorified the cluster algebra—our aim is to extend this to more general positroid varieties.

1 A non-commutative algebra ˛ ˛ ˛ ˛ ˛ ˛ ˛ 7 ˛ ˛ ˛ ˛ ˛ ˛ ˛ The dimer model D gives Q D a 2 ˛ ˛ ˛ ˛ ˛ ˛ ˛ determined set of ‚ -cycles and ˝ -cycles ˛ (bounding faces). 6 ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ Thus, letting Z “ C rr t ss , we can ˛ 3 consider matrix factorisations on Q D : ˛ ˛ ˛ ˛ ˛ ˛ ˛ representations with free Z -modules ˛ ˛ ˛ ˛ ˛ ˛ ˛ at each vertex, all having the same fixed 5 rank, and in which each ‚ - and ˝ -cycle 4 ˛ ˛ ˛ ˛ ˛ ˛ ˛ acts by t . (When the rank is 1 , these are given by perfect matchings.) When D is connected as a graph (equivalently | Q D | is a topological disc) these are precisely the A D -modules free over Z , where A D is the C -algebra determined by the following relations on Q : Each non-boundary (green) arrow a can be completed to either a ‚ -cycle or a ˝ -cycle by unique paths p ‚ a and p ˝ a ; we impose each relation p ‚ a “ p ˝ a . This is an example of a frozen Jacobian algebra , for the potential W “ ř p‚ -cycles q ´ ř p˝ -cycles q .

1 A non-commutative algebra ˛ ˛ ˛ ˛ ˛ ˛ ˛ 7 ˛ ˛ ˛ ˛ ˛ ˛ ˛ The dimer model D gives Q D a 2 ˛ ˛ ˛ ˛ ˛ ˛ ˛ determined set of ‚ -cycles and ˝ -cycles ˛ (bounding faces). 6 ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ Thus, letting Z “ C rr t ss , we can ˛ 3 consider matrix factorisations on Q D : ˛ ˛ ˛ ˛ ˛ ˛ ˛ representations with free Z -modules ˛ ˛ ˛ ˛ ˛ ˛ ˛ at each vertex, all having the same fixed 5 rank, and in which each ‚ - and ˝ -cycle 4 ˛ ˛ ˛ ˛ ˛ ˛ ˛ acts by t . (When the rank is 1 , these are given by perfect matchings.) When D is connected as a graph (equivalently | Q D | is a topological disc) these are precisely the A D -modules free over Z , where A D is the C -algebra determined by the following relations on Q : Each non-boundary (green) arrow a can be completed to either a ‚ -cycle or a ˝ -cycle by unique paths p ‚ a and p ˝ a ; we impose each relation p ‚ a “ p ˝ a . This is an example of a frozen Jacobian algebra , for the potential W “ ř p‚ -cycles q ´ ř p˝ -cycles q .

The boundary algebra Let e “ e 2 P A D be the sum of vertex idempotents at boundary vertices, and write B D “ eA D e for the boundary algebra . Theorem (Jensen–King–Su, Baur–King–Marsh) When σ D is the uniform permutation, the category GP p B D q “ t X P mod B D : Ext ą 0 B D p X, B D q “ 0 u categorifies the cluster algebra A D . Note: this presentation is historically backwards. In practice, Jensen–King–Su proved the above theorem for an explicitly defined ‘circle algebra’ C k , depending only on the chirality, which Baur–King–Marsh (slightly) later showed is isomorphic to B D whenever σ D is the uniform permutation. With hindsight, we can try to repeat this trick, this time using the boundary algebra description of B D as the (now more general) definition.

Categorification Theorem Let D be a connected consistent dimer model in the disc, with dimer algebra A “ A D and boundary algebra B “ B D . Then (1) B is Iwanaga–Gorenstein of Gorenstein dimension ď 3 ; that is, B is Noetherian and injdim B B, injdim B B ď 3 . In particular GP p B q is a Frobenius category. (2) The stable category GP p B q “ GP p B q{ proj B is a 2 -Calabi–Yau triangulated category. (3) A “ End B p eA q op and eA P GP p B q is cluster-tilting , that is add p eA q “ t X P GP p B q : Ext 1 B p X, eA q “ 0 u . This theorem follows from the following facts about the pair p A, e q : (1) A is Noetherian, (2) A { AeA is finite-dimensional, and (3) A is internally bimodule 3 -Calabi–Yau with respect to e .