Combinatorics of cluster structures in Schubert varieties

Transcription

1 Combinatorics of cluster structures in Schubert varieties arxiv: Slides available at M. Sherman-Bennett (UC Berkeley) joint work with K. Serhiyenko and L. Williams AMS Spring Eastern Sectional Meeting 2019 M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

2 The set-up Fix integers 0 < k < n. Gr k,n := {V C n : dim(v ) = k} M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

3 The set-up Fix integers 0 < k < n. Gr k,n := {V C n : dim(v ) = k} V Gr k,n full rank k n matrix A whose rows span V M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

4 The set-up Fix integers 0 < k < n. Gr k,n := {V C n : dim(v ) = k} V Gr k,n full rank k n matrix A whose rows span V I {1,..., n} with I = k. The Plücker coordinate P I (V ) is the maximal minor of A located in column set I. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

5 The set-up Fix integers 0 < k < n. Gr k,n := {V C n : dim(v ) = k} V Gr k,n full rank k n matrix A whose rows span V I {1,..., n} with I = k. The Plücker coordinate P I (V ) is the maximal minor of A located in column set I. The Schubert cell Ω I := {V Gr k,n : P I (V ) 0, P J (V ) = 0 for J < I } The open Schubert variety X I := Ω I \ {V Ω I : P I P I2 P In = 0} M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

6 The set-up Fix integers 0 < k < n. Gr k,n := {V C n : dim(v ) = k} V Gr k,n full rank k n matrix A whose rows span V I {1,..., n} with I = k. The Plücker coordinate P I (V ) is the maximal minor of A located in column set I. The Schubert cell Ω I := {V Gr k,n : P I (V ) 0, P J (V ) = 0 for J < I } The open Schubert variety XI := Ω I \ {V Ω I : P I P I2 P In = 0} Cluster algebra convention: Given a seed (x, Q) with x r+1,..., x N frozen, A(x, Q) is the C[x r+1 ±1,..., x ±1 N ]-algebra generated by the cluster variables. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

7 Motivation Theorem (Scott 06) C[Ĝr k,n ] is a cluster algebra with seeds (consisting entirely of Plücker coordinates) given by Postnikov s plabic graphs for Gr k,n. (Ĝr k,n is the affine cone over Gr k,n wrt Plücker embedding.) M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

8 Motivation Theorem (Scott 06) C[Ĝr k,n ] is a cluster algebra with seeds (consisting entirely of Plücker coordinates) given by Postnikov s plabic graphs for Gr k,n. (Ĝr k,n is the affine cone over Gr k,n wrt Plücker embedding.) Conjecture (Muller Speyer 16) Scott s result holds if you replace Gr k,n with an open positroid variety π k (R v,w ). M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

9 Main result Theorem (SSW 19) C[ X I ] is a cluster algebra, with seeds (consisting entirely of Plücker coordinates) given by plabic graphs for X I. ( X I is the affine cone over X I wrt Plücker embedding.) M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

10 Main result Theorem (SSW 19) C[ X I ] is a cluster algebra, with seeds (consisting entirely of Plücker coordinates) given by plabic graphs for X ( X I is the affine cone over X I I. wrt Plücker embedding.) More general result for open skew Schubert varieties π k (R v,xv ), where seeds for the cluster structure are given by generalized plabic graphs. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

11 Main result Theorem (SSW 19) C[ X I ] is a cluster algebra, with seeds (consisting entirely of Plücker coordinates) given by plabic graphs for X ( X I is the affine cone over X I I. wrt Plücker embedding.) More general result for open skew Schubert varieties π k (R v,xv ), where seeds for the cluster structure are given by generalized plabic graphs. We use a result of (Leclerc 16), who shows that coordinate rings of certain open Richardson varieties in the complete flag variety are cluster algebras. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

12 Postnikov s plabic graphs Friday, April 5, :19 AM A plabic graph of type (k, n) is a planar graph embedded in a disk with n boundary vertices labeled 1,..., n clockwise. Internal vertices colored white and black. Boundary vertices are adjacent to a unique internal vertex. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

13 Postnikov s plabic graphs Friday, April 5, :19 AM A plabic graph of type (k, n) is a planar graph embedded in a disk with n boundary vertices labeled 1,..., n clockwise. Internal vertices colored white and black. Boundary vertices are adjacent to a unique internal vertex. For generalized plabic graphs, we drop the condition that boundary vertices are labeled 1,..., n clockwise. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

14 Quivers from plabic graphs Figures for Schub_AMS Friday, April 5, :19 AM Let G be a reduced plabic graph of type (k, n). The dual quiver Q(G) is obtained by 1 Put a frozen vertex in each boundary face of G and a mutable vertex in each internal face. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

15 Figures for Schub_AMS Friday, Quivers April 5, 2019 from 9:19 plabic AM graphs Let G be a reduced plabic graph of type (k, n). The dual quiver Q(G) is obtained by 1 Put a frozen vertex in each boundary face of G and a mutable vertex in each internal face. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

16 Figures for Schub_AMS Friday, Quivers April 5, 2019 from 9:19 plabic AM graphs Let G be a reduced plabic graph of type (k, n). The dual quiver Q(G) is obtained by 1 Put a frozen vertex in each boundary face of G and a mutable vertex in each internal face. 2 Add arrows across properly colored edges so you see white vertex on the left. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

17 Figures for Schub_AMS Friday, April 5, :19 AM Quivers from plabic graphs Let G be a reduced plabic graph of type (k, n). The dual quiver Q(G) is obtained by 1 Put a frozen vertex in each boundary face of G and a mutable vertex in each internal face. 2 Add arrows across properly colored edges so you see white vertex on the left. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

18 Variables from plabic graphs A trip in G is a walk from boundary vertex to boundary vertex that turns maximally left at white vertices turns maximally right at black vertices M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

19 Variables from plabic graphs A trip in G is a walk from boundary vertex to boundary vertex that turns maximally left at white vertices turns maximally right at black vertices M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

20 Variables from plabic graphs A trip in G is a walk from boundary vertex to boundary vertex that turns maximally left at white vertices turns maximally right at black vertices Aside: The trip permutation of this graph is M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

21 Face labels If the trip T ends at j, put a j in all faces of G to the left of T. Do this for all trips. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

22 Face labels If the trip T ends at j, put a j in all faces of G to the left of T. Do this for all trips. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

23 Face labels If the trip T ends at j, put a j in all faces of G to the left of T. Do this for all trips. Fact: (Postnikov 06) All faces of G will be labeled by subsets of the same size (which is k). To get cluster variables, we interpret each face label as a Plücker coordinate. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

24 Which positroid is it for? Each reduced plabic graph corresponds to a unique positroid variety, determined by its trip permutation. The plabic graphs for XI have trip permutation π I = j 1 j 2... j n k i 1 i 2... i k where I = {i 1 < i 2 < < i k } and {1,..., n} \ I = {j 1 < j 2 < < j n k }. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

25 So in the end... The trip permutation of G is 24513, so this is a seed for X {1,3}. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

26 Applications Theorem Let G be a reduced plabic graph corresponding to XI, and let (x, Q(G)) be the associated seed. Then A(x, Q(G)) = C[ X I ]. Corollaries: Classification of when A(x, Q(G)) is finite type M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

27 Applications Theorem Let G be a reduced plabic graph corresponding to XI, and let (x, Q(G)) be the associated seed. Then A(x, Q(G)) = C[ X I ]. Corollaries: Classification of when A(x, Q(G)) is finite type From (Muller 13) and (Muller-Speyer 16), A(x, Q(G)) is locally acyclic, so it s locally a complete intersection and equal to its upper cluster algebra M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

28 Applications Theorem Let G be a reduced plabic graph corresponding to XI, and let (x, Q(G)) be the associated seed. Then A(x, Q(G)) = C[ X I ]. Corollaries: Classification of when A(x, Q(G)) is finite type From (Muller 13) and (Muller-Speyer 16), A(x, Q(G)) is locally acyclic, so it s locally a complete intersection and equal to its upper cluster algebra From (Ford-Serhiyenko 18), A(x, Q(G)) has green-to-red sequence, so satisfies the EGM property of (GHKK 18) and has a canonical basis of θ-functions parameterized by g-vectors. M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

29 More questions What about the other positroid varieties? M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

30 More questions What about the other positroid varieties? How do seeds/cluster structure from generalized plabic graphs compare to seeds/cluster structure from normal plabic graphs? M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12

31 More questions What about the other positroid varieties? How do seeds/cluster structure from generalized plabic graphs compare to seeds/cluster structure from normal plabic graphs? Is there a combinatorial characterization of compatibility of Plückers for Schubert, skew Schubert? (In the Schubert case, there are seeds in A(x, Q(G)) that consist entirely of Plücker coordinates, but the coordinates are not weakly separated.) M. Sherman-Bennett (UC Berkeley) Schubert cluster structure AMS Spring Sectional / 12