Dimer models and cluster categories of Grassmannians Karin Baur - - PowerPoint PPT Presentation

Dimer models and cluster categories of Grassmannians

Karin Baur

University of Graz

Rome, October 18, 2016

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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

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Coordinate ring of Grassmannian

Grk,n = {k-spaces in Cn} ∋ pt → (v1, . . . , vn) with vi ∈ Ck. full rank k × n-matrix /GLk.

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Coordinate ring of Grassmannian

Grk,n = {k-spaces in Cn} ∋ pt → (v1, . . . , vn) with vi ∈ Ck. full rank k × n-matrix /GLk. For I = {1 ≤ i1 < i2 < · · · < ik ≤ n}: ∆I := det(vi1, vi2, . . . , vik) The I-th Pl¨ ucker coordinate (up to C∗-multiplication).

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Coordinate ring of Grassmannian

Grk,n = {k-spaces in Cn} ∋ pt → (v1, . . . , vn) with vi ∈ Ck. full rank k × n-matrix /GLk. For I = {1 ≤ i1 < i2 < · · · < ik ≤ n}: ∆I := det(vi1, vi2, . . . , vik) The I-th Pl¨ ucker coordinate (up to C∗-multiplication). The Pl¨ ucker coordinates generate C[ Grk,n] (in deg 1). They satisfy the Pl¨ ucker relations (deg 2 relations).

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Cluster algebra structure of C[ Grk,n]

Theorem (Fomin-Zelevinsky, Scott)

D a (k, n)-diagram. X(D) := {∆I(R) | R alternating region of D}. = ⇒ every element of C[ Grk,n] is a Laurent polynomial in X(D).

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Cluster algebra structure of C[ Grk,n]

Theorem (Fomin-Zelevinsky, Scott)

D a (k, n)-diagram. X(D) := {∆I(R) | R alternating region of D}. = ⇒ every element of C[ Grk,n] is a Laurent polynomial in X(D). X(D) is a cluster, C[ Grk,n] a cluster algebra. Exchange relations: Pl¨ ucker relations

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Cluster algebra structure of C[ Grk,n]

Theorem (Fomin-Zelevinsky, Scott)

D a (k, n)-diagram. X(D) := {∆I(R) | R alternating region of D}. = ⇒ every element of C[ Grk,n] is a Laurent polynomial in X(D). X(D) is a cluster, C[ Grk,n] a cluster algebra. Exchange relations: Pl¨ ucker relations

Proofs

Fomin-Zelevinsky k = 2 (triangulations!). Scott: arbitrary k (alternating strand diagrams).

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Construction of cluster categories

Cluster categories (type An)

Let Q be a quiver of Dynkin type An. CQ path algebra of Q

1 2 3 α β

{e1, e2, e3, α, β, β ◦ α}

CQ-mod: category of CQ-modules

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Construction of cluster categories

Cluster categories (type An)

Let Q be a quiver of Dynkin type An. CQ path algebra of Q

1 2 3 α β

{e1, e2, e3, α, β, β ◦ α}

CQ-mod: category of CQ-modules Cluster category C(Q) :=Db(CQ)/τ −1[1]

[Buan-Marsh-Reineke-Reiten-Todorov ’05, Caldero-Chapoton-Schiffler ’05] 5 / 17

Construction of cluster categories

Cluster categories (type An)

Let Q be a quiver of Dynkin type An. CQ path algebra of Q

1 2 3 α β

{e1, e2, e3, α, β, β ◦ α}

CQ-mod: category of CQ-modules Cluster category C(Q) :=Db(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 An. Intrinsic construction?

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(k, n) - diagrams

Alternating strand diagrams (Postnikov ’06), on disk (surfaces). n marked points on boundary, {1, 2, . . . , n}, clockwise Si, i = 1, . . . , n oriented strands, Si : i → i + k (reduce mod n)

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(k, n) - diagrams

Alternating strand diagrams (Postnikov ’06), on disk (surfaces). n marked points on boundary, {1, 2, . . . , n}, clockwise Si, i = 1, . . . , n oriented strands, Si : 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:

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Example of a (3, 7)-diagram

1 2 3 4 5 6 7

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Example of a (3, 7)-diagram

1 2 3 4 5 6 7

123 234 345 456 567 167 127 245 145 156 157 147 124

Alternating regions. Label i if to the left of Si. Always k labels.

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Dimer models

Definition (dimer model with boundary)

A (finite, oriented) dimer model with boundary is Q = (Q0, Q1, Q2) with

• 1. Q2 = Q+

2 ⊔ Q− 2 faces, ∂ : Q2 → Qcyc, F → ∂F

• 2. Arrows have face mult. 2 or 1: internal or boundary arrows.
• 3. arrows at each vertex alternate “in”/“out”

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Dimer models

Definition (dimer model with boundary)

A (finite, oriented) dimer model with boundary is Q = (Q0, Q1, Q2) with

• 1. Q2 = Q+

2 ⊔ Q− 2 faces, ∂ : Q2 → Qcyc, 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.

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

123 234 345 456 567 167 127 245 145 156 157 147 124 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.

123 234 345 456 567 167 127 245 145 156 157 147 124 123 234 345 456 567 167 127 245 145 156 157 147 124 9 / 17

Dimer algebras

Definition (dimer algebra)

Q dimer model w boundary. The dimer algebra of Q is ΛQ := CQ/∂W . W : natural potential on Q, W = WQ :=

• F∈Q+

2

F −

• F∈Q−

2

F ∂W : cyclic derivatives wrt internal arrows only.

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Dimer algebras

Definition (dimer algebra)

Q dimer model w boundary. The dimer algebra of Q is ΛQ := CQ/∂W . W : natural potential on Q, W = WQ :=

• F∈Q+

2

F −

• F∈Q−

2

F ∂W : cyclic derivatives wrt internal arrows only. α an arrow in F1 and in F2. Two cycles p1 ◦ α and p2 ◦ α.

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Dimer algebras

Definition (dimer algebra)

Q dimer model w boundary. The dimer algebra of Q is ΛQ := CQ/∂W . W : natural potential on Q, W = WQ :=

• F∈Q+

2

F −

• F∈Q−

2

F ∂W : cyclic derivatives wrt internal arrows only. α an arrow in F1 and in F2. Two cycles p1 ◦ α and p2 ◦ α. ∂W /(∂α) : p1 = p2.

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... and their boundary

Q dimer model w boundary. ΛQ = CQ/∂W the dimer algebra of Q.

Definition (boundary algebra of Q)

Let eb be the sum of the boundary idempotents of kQ. Then we define the boundary algebra of Q as BQ := ebΛQeb

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Module category with Grassmannian structure

JKS-algebra

[Jensen-King-Su]

Γn: vertices 1, 2, . . . , n, arrows: xi : i − 1 → i, yi : i → i − 1.

1 2 3 4 5 6 x1 x2 x3 x4 x5 x6 y6 y1 y2 y3 y4 y5

B := Bk,n := CΓn/(rel’s) (rel’s): “xy = yx”, “xk = y n−k”.

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Module category with Grassmannian structure

JKS-algebra

[Jensen-King-Su]

Γn: vertices 1, 2, . . . , n, arrows: xi : i − 1 → i, yi : i → i − 1.

1 2 3 4 5 6 x1 x2 x3 x4 x5 x6 y6 y1 y2 y3 y4 y5

B := Bk,n := CΓn/(rel’s) (rel’s): “xy = yx”, “xk = y n−k”. t := xiyi is central in B. Centre of B is Z = C[t].

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Module category with Grassmannian structure

JKS-algebra

[Jensen-King-Su]

Γn: vertices 1, 2, . . . , n, arrows: xi : i − 1 → i, yi : i → i − 1.

1 2 3 4 5 6 x1 x2 x3 x4 x5 x6 y6 y1 y2 y3 y4 y5

B := Bk,n := CΓn/(rel’s) (rel’s): “xy = yx”, “xk = y n−k”. t := xiyi is central in B. Centre of B is Z = C[t].

Frobenius category

F = Fk,n := CM(Bk,n) = {M | M free over Z} max. CM modules.

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Module category with Grassmannian structure

JKS-algebra

[Jensen-King-Su]

Γn: vertices 1, 2, . . . , n, arrows: xi : i − 1 → i, yi : i → i − 1.

1 2 3 4 5 6 x1 x2 x3 x4 x5 x6 y6 y1 y2 y3 y4 y5

B := Bk,n := CΓn/(rel’s) (rel’s): “xy = yx”, “xk = y n−k”. t := xiyi is central in B. Centre of B is Z = C[t].

Frobenius category

F = Fk,n := CM(Bk,n) = {M | M free over Z} max. CM modules. M ∈ F: collection of copies of Z, linked via xi, yi, on a cylinder.

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Rank one modules

MI for I = {1, 4, 5}. Infinite dimensional. Rim.

7 7 2 1 x1 x4 x5 y2 y3 y7 y6 6

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Properties of F

Properties (Jensen-King-Su, B-Bogdanic)

◮ F is Frobenius =

⇒ F triangulated;

◮ rk 1 indecomposables in bijection with k-subsets; ◮ Ext1(MI, MJ) = 0

iff I and J don’t cross;

◮ T := I∈D MI is maximal rigid in F; so F a cluster category.

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Properties of F

Properties (Jensen-King-Su, B-Bogdanic)

◮ F is Frobenius =

⇒ F triangulated;

◮ rk 1 indecomposables in bijection with k-subsets; ◮ Ext1(MI, MJ) = 0

iff I and J don’t cross;

◮ T := I∈D MI is maximal rigid in F; so F a cluster category. ◮ periodic resolutions in F, period divides 2n, ◮ I, J crossing:

Ext2m+1(MI, MJ) = C[t]/(ta1) × · · · × C[t]/(tar ) Ext2m(MI, MJ) = C[t]/(ta)

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F dimer algebra

D a (k, n)-diagram. Q = Q(D) the associated dimer. ΛQ = C/∂W dimer algebra. B, F as before. TD :=

I∈Q(D) MI ∈ F is maximal rigid.

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F dimer algebra

D a (k, n)-diagram. Q = Q(D) the associated dimer. ΛQ = C/∂W dimer algebra. B, F as before. TD :=

I∈Q(D) MI ∈ F is maximal rigid.

Theorem (B-King-Marsh)

• 1. ΛQ ∼

= EndB(TD).

• 2. BQ = ebΛQeb ∼

= Bop

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F dimer algebra

D a (k, n)-diagram. Q = Q(D) the associated dimer. ΛQ = C/∂W dimer algebra. B, F as before. TD :=

I∈Q(D) MI ∈ F is maximal rigid.

Theorem (B-King-Marsh)

• 1. ΛQ ∼

= EndB(TD).

• 2. BQ = ebΛQeb ∼

= Bop

Corollary

The boundary algebra BQ (of Q) independent of the choice of D.

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Further work

◮ k = 2: Result (2) for arbitrary surface w boundary (no punct) ◮ versions with punctures (relax strand diagram notion) ◮ B = Bk,n (Gorenstein, centre C[t]). Infinite global dimension.

Take A =EndB(T)op instead : finite global dimension.

◮ Strand diagrams from tilings (B-Martin).

(n − 1, 2n)-diagrams from tilings (B-Martin). From GLm-webs (Andritsch). Boundary algebras for m > 2?

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Bibliography

◮ Andritsch, The boundary algebra of a GL2-web, master’s thesis,

2015.

◮ B-Bogdanic, Extensions between Cohen-Macaulay modules of

Grassmannian cluster categories, arXiv:1601.05943

◮ B-King-Marsh, Dimer models and cluster categories of

Grassmannians, Proc. LMS 2016.

◮ B-Martin, The fibres of the Scott map on polygon tilings are the flip

equivalence classes, arXiv:1601.06080

◮ Jensen-King-Su, A categorification of Grassmannian cluster

algebras, Proc. LMS 2016.

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