dimer models and cluster categories of grassmanniansconf/ltca16/rome conference...

Dimer models and cluster categories ofGrassmannians

Karin Baur

University of Graz

Rome, October 18, 2016

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MotivationCluster algebra structure of GrassmanniansConstruction of cluster categories

(k , n) - diagramsDefinitionExample

Dimer models and dimer algebrasDimer modelsDimer algebras

Module category with Grassmannian structureAn algebra of preprojective typeProperties of F

F ! dimer algebraBack to the dimer algebra

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Coordinate ring of Grassmannian

Grk,n = {k-spaces in Cn} ∋ pt 7→ (v1, . . . , vn) with vi ∈ Ck .

full rank k × n-matrix /GLk .

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For I = {1 ≤ i1 < i2 < · · · < ik ≤ n}: ∆I := det(vi1 , vi2 , . . . , vik )

The I -th Plucker coordinate (up to C∗-multiplication).

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For I = {1 ≤ i1 < i2 < · · · < ik ≤ n}: ∆I := det(vi1 , vi2 , . . . , vik )

The I -th Plucker coordinate (up to C∗-multiplication).

The Plucker coordinates generate C[Grk,n] (in deg 1).

They satisfy the Plucker 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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X (D) is a cluster, C[Grk,n] a cluster algebra.Exchange relations: Plucker relations

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X (D) is a cluster, C[Grk,n] a cluster algebra.Exchange relations: Plucker relations

ProofsFomin-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 3α β {e1, e2, e3, α, β, β ◦ α}

CQ-mod: category of CQ-modules

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1 2 3α β {e1, e2, e3, α, β, β ◦ α}


Cluster category C(Q) :=Db(CQ)/τ−1[1]

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

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1 2 3α β {e1, e2, e3, α, β, β ◦ α}


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 7→ 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 7→ 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

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Example of a (3, 7)-diagram

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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 isQ = (Q0,Q1,Q2) with

1. Q2 = Q+2 ⊔Q−

2 faces, ∂ : Q2 → Qcyc, F 7→ ∂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 isQ = (Q0,Q1,Q2) with

1. Q2 = Q+2 ⊔Q−

2 faces, ∂ : Q2 → Qcyc, F 7→ ∂F

2. Arrows have face mult. 2 or 1: internal or boundary arrows.

3. arrows at each vertex alternate “in”/“out”

RemarkQ 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.

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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.

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Dimer algebras

Definition (dimer algebra)

Q dimer model w boundary. The dimer algebra of Q isΛQ := 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




W = WQ :=∑

F∈Q+2

F −∑

F∈Q−

2

F


α an arrow in F1 and in F2. Two cycles p1 ◦ α and p2 ◦ α.

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Dimer algebras




W = WQ :=∑

F∈Q+2

F −∑

F∈Q−

2

F


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 wedefine 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.

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y5B := Bk,n := CΓn/(rel’s)(rel’s): “xy = yx”,“xk = yn−k”.

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y5B := Bk,n := CΓn/(rel’s)(rel’s): “xy = yx”,“xk = yn−k”.t :=

∑xiyi is central in B .

Centre of B is Z = C[t].

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Frobenius category

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

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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.

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1

x1

x4

x5y2

y3 y7

y6

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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




Theorem (B-King-Marsh)

1. ΛQ∼= EndB(TD).

2. BQ = ebΛQeb ∼= Bop

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F ! dimer algebra




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 ofGrassmannian cluster categories, arXiv:1601.05943

◮ B-King-Marsh, Dimer models and cluster categories ofGrassmannians, Proc. LMS 2016.

◮ B-Martin, The fibres of the Scott map on polygon tilings are the flipequivalence classes, arXiv:1601.06080

◮ Jensen-King-Su, A categorification of Grassmannian clusteralgebras, Proc. LMS 2016.

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dimer models and cluster categories of grassmanniansconf/ltca16/rome conference...

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