Dimer models and cluster categories of Grassmannians

Transcription

1 Dimer models and cluster categories of Grassmannians Karin Baur University of Graz Rome, October 18, /17

2 Motivation Cluster algebra structure of Grassmannians Construction of cluster categories (k,n) - diagrams Definition Example Dimer models and dimer algebras Dimer models Dimer algebras Module category with Grassmannian structure An algebra of preprojective type Properties of F F dimer algebra Back to the dimer algebra 2/17

3 Coordinate ring of Grassmannian Gr k,n = {k-spaces in C n } pt (v 1,...,v n ) with v i C k. full rank k n-matrix /GL k. 3/17

4 Coordinate ring of Grassmannian Gr k,n = {k-spaces in C n } pt (v 1,...,v n ) with v i C k. full rank k n-matrix /GL k. For I = {1 i 1 < i 2 < < i k n}: I := det(v i1,v i2,...,v ik ) The I-th Plücker coordinate (up to C -multiplication). 3/17

5 Coordinate ring of Grassmannian Gr k,n = {k-spaces in C n } pt (v 1,...,v n ) with v i C k. full rank k n-matrix /GL k. For I = {1 i 1 < i 2 < < i k n}: I := det(v i1,v i2,...,v ik ) The I-th Plücker coordinate (up to C -multiplication). The Plücker coordinates generate C[Ĝr k,n ] (in deg 1). They satisfy the Plücker relations (deg 2 relations). 3/17

6 Cluster algebra structure of C[Ĝr k,n ] Theorem (Fomin-Zelevinsky, Scott) D a (k,n)-diagram. X(D) := { I(R) R alternating region of D}. = every element of C[Ĝr k,n ] is a Laurent polynomial in X(D). 4/17

7 Cluster algebra structure of C[Ĝr k,n ] Theorem (Fomin-Zelevinsky, Scott) D a (k,n)-diagram. X(D) := { I(R) R alternating region of D}. = every element of C[Ĝr k,n ] is a Laurent polynomial in X(D). X(D) is a cluster, C[Ĝr k,n ] a cluster algebra. Exchange relations: Plücker relations 4/17

8 Cluster algebra structure of C[Ĝr k,n ] Theorem (Fomin-Zelevinsky, Scott) D a (k,n)-diagram. X(D) := { I(R) R alternating region of D}. = every element of C[Ĝr k,n ] is a Laurent polynomial in X(D). X(D) is a cluster, C[Ĝr k,n ] a cluster algebra. Exchange relations: Plücker relations Proofs Fomin-Zelevinsky k = 2 (triangulations!). Scott: arbitrary k (alternating strand diagrams). 4/17

9 Construction of cluster categories Cluster categories (type A n ) Let Q be a quiver of Dynkin type A n. CQ path algebra of Q α β {e 1,e 2,e 3,α,β,β α} CQ-mod: category of CQ-modules 5/17

10 Construction of cluster categories Cluster categories (type A n ) Let Q be a quiver of Dynkin type A n. CQ path algebra of Q α β {e 1,e 2,e 3,α,β,β α} CQ-mod: category of CQ-modules Cluster category C(Q) :=D b (CQ)/τ 1 [1] [Buan-Marsh-Reineke-Reiten-Todorov 05, Caldero-Chapoton-Schiffler 05] 5/17

11 Construction of cluster categories Cluster categories (type A n ) Let Q be a quiver of Dynkin type A n. CQ path algebra of Q α β {e 1,e 2,e 3,α,β,β α} CQ-mod: category of CQ-modules Cluster category C(Q) :=D b (CQ)/τ 1 [1] [Buan-Marsh-Reineke-Reiten-Todorov 05, Caldero-Chapoton-Schiffler 05] C(Q) equiv to C(Q ) for Q and Q different orientations of A n. Intrinsic construction? 5/17

12 (k,n) - diagrams Alternating strand diagrams (Postnikov 06), on disk (surfaces). n marked points on boundary, {1,2,...,n}, clockwise S i, i = 1,...,n oriented strands, S i : i i +k (reduce mod n) 6/17

13 (k,n) - diagrams Alternating strand diagrams (Postnikov 06), on disk (surfaces). n marked points on boundary, {1,2,...,n}, clockwise S i, i = 1,...,n oriented strands, S i : i i +k (reduce mod n) Rules crossings alternate, multiplicity 2, transversal no un-oriented lenses, no self-crossings up to isotopy fixing endpoints, up to two equivalences: 6/17

14 Example of a (3, 7)-diagram /17

15 Example of a (3, 7)-diagram Alternating regions. Label i if to the left of S i. Always k labels. 7/17

16 Dimer models Definition (dimer model with boundary) A (finite, oriented) dimer model with boundary is Q = (Q 0,Q 1,Q 2 ) with 1. Q 2 = Q + 2 Q 2 faces, : Q 2 Q cyc, F F 2. Arrows have face mult. 2 or 1: internal or boundary arrows. 3. arrows at each vertex alternate in / out 8/17

17 Dimer models Definition (dimer model with boundary) A (finite, oriented) dimer model with boundary is Q = (Q 0,Q 1,Q 2 ) with 1. Q 2 = Q + 2 Q 2 faces, : Q 2 Q cyc, F F 2. Arrows have face mult. 2 or 1: internal or boundary arrows. 3. arrows at each vertex alternate in / out Remark Q as above oriented surface Q with boundary. Source for dimer models: (k, n)-diagrams. 8/17

18 D a (k,n)-diagram Q(D) a dimer with boundary: k-subsets: Q(D) 0. Arrows: flow. Faces: oriented regions in D /17

19 D a (k,n)-diagram Q(D) a dimer with boundary: k-subsets: Q(D) 0. Arrows: flow. Faces: oriented regions in D /17

20 Dimer algebras Definition (dimer algebra) Q dimer model w boundary. The dimer algebra of Q is Λ Q := CQ/ W. W: natural potential on Q, W = W Q := F F F Q + 2 F Q 2 W: cyclic derivatives wrt internal arrows only. 10/17

21 Dimer algebras Definition (dimer algebra) Q dimer model w boundary. The dimer algebra of Q is Λ Q := CQ/ W. W: natural potential on Q, W = W Q := F F F Q + 2 F Q 2 W: cyclic derivatives wrt internal arrows only. α an arrow in F 1 and in F 2. Two cycles p 1 α and p 2 α. 10/17

22 Dimer algebras Definition (dimer algebra) Q dimer model w boundary. The dimer algebra of Q is Λ Q := CQ/ W. W: natural potential on Q, W = W Q := F F F Q + 2 F Q 2 W: cyclic derivatives wrt internal arrows only. α an arrow in F 1 and in F 2. Two cycles p 1 α and p 2 α. W/( α) : p 1 = p 2. 10/17

23 ... and their boundary Q dimer model w boundary. Λ Q = CQ/ W the dimer algebra of Q. Definition (boundary algebra of Q) Let e b be the sum of the boundary idempotents of kq. Then we define the boundary algebra of Q as B Q := e b Λ Q e b 11/17

24 Module category with Grassmannian structure JKS-algebra [Jensen-King-Su] Γ n : vertices 1,2,...,n, arrows: x i : i 1 i, y i : i i 1. x6 6 y6 1 x5 5 y5 y1 x1 2 B := B k,n := CΓ n /(rel s) (rel s): xy = yx, x k = y n k. y4 y2 x4 x2 4 y3 3 x3 12/17

25 Module category with Grassmannian structure JKS-algebra [Jensen-King-Su] Γ n : vertices 1,2,...,n, arrows: x i : i 1 i, y i : i i 1. x6 6 y6 1 x5 5 x4 y5 y4 y1 y2 x1 2 x2 B := B k,n := CΓ n /(rel s) (rel s): xy = yx, x k = y n k. t := x i y i is central in B. Centre of B is Z = C[t]. 4 y3 3 x3 12/17

26 Module category with Grassmannian structure JKS-algebra [Jensen-King-Su] Γ n : vertices 1,2,...,n, arrows: x i : i 1 i, y i : i i 1. x6 6 y6 1 x5 5 x4 y5 y4 y1 y2 x1 2 x2 B := B k,n := CΓ n /(rel s) (rel s): xy = yx, x k = y n k. t := x i y i is central in B. Centre of B is Z = C[t]. 4 y3 3 x3 Frobenius category F = F k,n := CM(B k,n ) = {M M free over Z} max. CM modules. 12/17

27 Module category with Grassmannian structure JKS-algebra [Jensen-King-Su] Γ n : vertices 1,2,...,n, arrows: x i : i 1 i, y i : i i 1. x6 6 y6 1 x5 5 x4 y5 y4 y1 y2 x1 2 x2 B := B k,n := CΓ n /(rel s) (rel s): xy = yx, x k = y n k. t := x i y i is central in B. Centre of B is Z = C[t]. 4 y3 3 x3 Frobenius category F = F k,n := CM(B k,n ) = {M M free over Z} max. CM modules. M F: collection of copies of Z, linked via x i, y i, on a cylinder. 12/17

28 Rank one modules M I for I = {1,4,5}. Infinite dimensional. Rim y 3 x 4 y 7 x 1 1 y 2 x 5 y 6 13/17

29 Properties of F Properties (Jensen-King-Su, B-Bogdanic) F is Frobenius = F triangulated; rk 1 indecomposables in bijection with k-subsets; Ext 1 (M I,M J ) = 0 iff I and J don t cross; T := I D M I is maximal rigid in F; so F a cluster category. 14/17

30 Properties of F Properties (Jensen-King-Su, B-Bogdanic) F is Frobenius = F triangulated; rk 1 indecomposables in bijection with k-subsets; Ext 1 (M I,M J ) = 0 iff I and J don t cross; T := I D M I is maximal rigid in F; so F a cluster category. periodic resolutions in F, period divides 2n, I, J crossing: Ext 2m+1 (M I,M J ) = C[t]/(t a 1 ) C[t]/(t ar ) Ext 2m (M I,M J ) = C[t]/(t a ) 14/17

31 F dimer algebra D a (k,n)-diagram. Q = Q(D) the associated dimer. Λ Q = C/ W dimer algebra. B, F as before. T D := I Q(D) M I F is maximal rigid. 15/17

32 F dimer algebra D a (k,n)-diagram. Q = Q(D) the associated dimer. Λ Q = C/ W dimer algebra. B, F as before. Theorem (B-King-Marsh) 1. Λ Q = EndB (T D ). 2. B Q = e b Λ Q e b = B op T D := I Q(D) M I F is maximal rigid. 15/17

33 F dimer algebra D a (k,n)-diagram. Q = Q(D) the associated dimer. Λ Q = C/ W dimer algebra. B, F as before. Theorem (B-King-Marsh) 1. Λ Q = EndB (T D ). 2. B Q = e b Λ Q e b = B op T D := I Q(D) M I F is maximal rigid. Corollary The boundary algebra B Q (of Q) independent of the choice of D. 15/17

34 Further work k = 2: Result (2) for arbitrary surface w boundary (no punct) versions with punctures (relax strand diagram notion) B = B k,n (Gorenstein, centre C[t]). Infinite global dimension. Take A =End B (T) op instead : finite global dimension. Strand diagrams from tilings (B-Martin). (n 1, 2n)-diagrams from tilings (B-Martin). From GL m -webs (Andritsch). Boundary algebras for m > 2? 16/17

35 Bibliography Andritsch, The boundary algebra of a GL 2 -web, master s thesis, B-Bogdanic, Extensions between Cohen-Macaulay modules of Grassmannian cluster categories, arxiv: B-King-Marsh, Dimer models and cluster categories of Grassmannians, Proc. LMS B-Martin, The fibres of the Scott map on polygon tilings are the flip equivalence classes, arxiv: Jensen-King-Su, A categorification of Grassmannian cluster algebras, Proc. LMS /17