Dimer models and cluster categories of Grassmannians Karin Baur University of Graz Rome, October 18, 2016 1 / 17
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
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
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 i 1 , v i 2 , . . . , v i k ) ucker coordinate (up to C ∗ -multiplication). The I -th Pl¨ 3 / 17
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 i 1 , v i 2 , . . . , v i k ) ucker coordinate (up to C ∗ -multiplication). The I -th Pl¨ ucker coordinates generate C [ � The Pl¨ Gr k , n ] (in deg 1). They satisfy the Pl¨ ucker relations (deg 2 relations). 3 / 17
Cluster algebra structure of C [ � Gr k , n ] Theorem (Fomin-Zelevinsky, Scott) D a ( k , n )-diagram. X ( D ) := { ∆ I ( R ) | R alternating region of D } . ⇒ every element of C [ � = Gr k , n ] is a Laurent polynomial in X ( D ). 4 / 17
Cluster algebra structure of C [ � Gr k , n ] Theorem (Fomin-Zelevinsky, Scott) D a ( k , n )-diagram. X ( D ) := { ∆ I ( R ) | R alternating region of D } . ⇒ every element of C [ � = Gr k , n ] is a Laurent polynomial in X ( D ). � X ( D ) is a cluster, C [ � Gr k , n ] a cluster algebra. Exchange relations: Pl¨ ucker relations 4 / 17
Cluster algebra structure of C [ � Gr k , n ] Theorem (Fomin-Zelevinsky, Scott) D a ( k , n )-diagram. X ( D ) := { ∆ I ( R ) | R alternating region of D } . ⇒ every element of C [ � = Gr k , n ] is a Laurent polynomial in X ( D ). � X ( D ) is a cluster, C [ � Gr k , n ] a cluster algebra. Exchange relations: Pl¨ ucker relations Proofs Fomin-Zelevinsky k = 2 (triangulations!). Scott: arbitrary k (alternating strand diagrams). 4 / 17
Construction of cluster categories Cluster categories (type A n ) Let Q be a quiver of Dynkin type A n . C Q path algebra of Q 1 2 3 α β { e 1 , e 2 , e 3 , α, β, β ◦ α } C Q -mod: category of C Q -modules 5 / 17
Construction of cluster categories Cluster categories (type A n ) Let Q be a quiver of Dynkin type A n . C Q path algebra of Q 1 2 3 α β { e 1 , e 2 , e 3 , α, β, β ◦ α } C Q -mod: category of C Q -modules Cluster category C ( Q ) :=D b ( C Q ) /τ − 1 [1] [Buan-Marsh-Reineke-Reiten-Todorov ’05, Caldero-Chapoton-Schiffler ’05] 5 / 17
Construction of cluster categories Cluster categories (type A n ) Let Q be a quiver of Dynkin type A n . C Q path algebra of Q 1 2 3 α β { e 1 , e 2 , e 3 , α, β, β ◦ α } C Q -mod: category of C Q -modules Cluster category C ( Q ) :=D b ( C Q ) /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
( 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
( 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
Example of a (3 , 7)-diagram 1 7 567 167 156 2 157 456 147 145 6 127 345 245 124 3 123 5 234 4 7 / 17
Example of a (3 , 7)-diagram 1 7 567 167 156 2 157 456 147 145 6 127 345 245 124 3 123 5 234 4 Alternating regions. Label i if to the left of S i . Always k labels. 7 / 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” 8 / 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
D a ( k , n )-diagram � Q ( D ) a dimer with boundary: k -subsets: Q ( D ) 0 . Arrows: “flow”. Faces: oriented regions in D . 567 167 156 157 456 147 145 127 345 245 124 123 234 9 / 17
D a ( k , n )-diagram � Q ( D ) a dimer with boundary: k -subsets: Q ( D ) 0 . Arrows: “flow”. Faces: oriented regions in D . 567 567 167 167 156 156 157 157 456 456 147 145 145 147 127 127 245 345 245 124 345 124 123 234 123 234 9 / 17
Dimer algebras Definition (dimer algebra) Q dimer model w boundary. The dimer algebra of Q is Λ Q := C Q /∂ W . W : natural potential on Q , � � W = W Q := F − F F ∈ Q + F ∈ Q − 2 2 ∂ W : cyclic derivatives wrt internal arrows only. 10 / 17
Dimer algebras Definition (dimer algebra) Q dimer model w boundary. The dimer algebra of Q is Λ Q := C Q /∂ W . W : natural potential on Q , � � W = W Q := F − F F ∈ Q + F ∈ Q − 2 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
Dimer algebras Definition (dimer algebra) Q dimer model w boundary. The dimer algebra of Q is Λ Q := C Q /∂ W . W : natural potential on Q , � � W = W Q := F − F F ∈ Q + F ∈ Q − 2 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
... and their boundary Q dimer model w boundary. Λ Q = C Q /∂ 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
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. x 6 6 y 6 1 B := B k , n := C Γ n / (rel’s) x 1 x 5 y 5 y 1 (rel’s): “ xy = yx ”, “ x k = y n − k ”. 5 2 y 4 y 2 x 4 x 2 4 y 3 3 x 3 12 / 17
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. x 6 6 y 6 1 B := B k , n := C Γ n / (rel’s) x 1 x 5 y 5 y 1 (rel’s): “ xy = yx ”, “ x k = y n − k ”. 5 2 t := � x i y i is central in B . y 4 y 2 Centre of B is Z = C [ t ]. x 4 x 2 4 y 3 3 x 3 12 / 17
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. x 6 6 y 6 1 B := B k , n := C Γ n / (rel’s) x 1 x 5 y 5 y 1 (rel’s): “ xy = yx ”, “ x k = y n − k ”. 5 2 t := � x i y i is central in B . y 4 y 2 Centre of B is Z = C [ t ]. x 4 x 2 4 y 3 3 x 3 Frobenius category F = F k , n := CM( B k , n ) = { M | M free over Z } max. CM modules. 12 / 17
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. x 6 6 y 6 1 B := B k , n := C Γ n / (rel’s) x 1 x 5 y 5 y 1 (rel’s): “ xy = yx ”, “ x k = y n − k ”. 5 2 t := � x i y i is central in B . y 4 y 2 Centre of B is Z = C [ t ]. x 4 x 2 4 y 3 3 x 3 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
Rank one modules M I for I = { 1 , 4 , 5 } . Infinite dimensional. Rim. 6 7 y 3 x 4 y 7 2 7 x 1 y 2 x 5 y 6 1 13 / 17
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
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