Dimer models on cylinders over Dynkin diagrams Maitreyee Kulkarni Conference on Geometric methods in Representation Theory Louisiana State University 1 / 24
Notation • G: simply connected complex algebraic group of type ADE • P: parabolic subgroup • G/P: partial flag variety • C [ G / P ]: coordinate ring 2 / 24
Overview Geiss - Leclerc - Schr¨ oer There exists a cluster algebra structure on C [ G / P ] using subcategory of modules over a preprojective algebra Jensen - King - Su C [ Gr ( k , n )] has a categorification via a category of Cohen-Macaulay modules of a certain ring. Baur - King - Marsh Gave a combinatorial description of the JKS categorification via dimer models. 3 / 24
Goal JKS: BKM: k-subset I of { 1 , 2 , . . . , n } CM module M I a vertex of a quiver a quiver or dimer model D T D = ⊕ I M I End B ( T D ) Jacobian algebra A D Theorem (BKM) The Jacobian algebra A D ∼ = End B T D . Want a combinatorial model for cluster structure of double Bruhat cells on Kac–Moody group G . 4 / 24
Jacobian algebra Definition a 1 2 Let Q : with potential P = abcd d b 3 4 c Cyclic derivatives, ∂ a ( P ) = bcd , ∂ b ( P ) = cda , ∂ c ( P ) = dab , ∂ d ( P ) = abc 5 / 24
Jacobian algebra Definition a 1 2 Let Q : with potential P = abcd d b 3 4 c Cyclic derivatives, ∂ a ( P ) = bcd , ∂ b ( P ) = cda , ∂ c ( P ) = dab , ∂ d ( P ) = abc Jacobian ideal, J ( P ) = Ideal generated by { ∂ a ( P ) , ∂ b ( P ) , ∂ c ( P ) , ∂ d ( P ) } Jacobian algebra, A ( Q , P ) = C Q / J ( P ) 6 / 24
Jacobian algebra Definition a 1 2 Let Q : with potential P = abcd d b 3 4 c Cyclic derivatives, ∂ a ( P ) = bcd , ∂ b ( P ) = cda , ∂ c ( P ) = dab , ∂ d ( P ) = abc Jacobian ideal, J ( P ) = Ideal generated by { ∂ a ( P ) , ∂ b ( P ) , ∂ c ( P ) , ∂ d ( P ) } Jacobian algebra, A ( Q , P ) = C Q / J ( P ) Superpotential S = � anticlockwise cycles - � clockwise cycles 7 / 24
Cluster structure on C [ G r ( k , n )] JKS: BKM: k-subset I of { 1 , 2 , . . . , n } CM module M I a vertex of a quiver T D = ⊕ I M I a quiver or dimer model D ∼ End B ( T D ) Jacobian algebra A D = Want a combinatorial model for cluster structure of double Bruhat cells on Kac–Moody group G . 8 / 24
Quivers from double Bruhat cells A Kac-Moody group G behaves like a semi-simple Lie group. Fact In particular, G is a disjoint union of the double Bruhat cells G u , v = B + uB + ∩ B − vB − where u , v ∈ W Berenstein, Fomin and Zelevinsky gave a combinatorial way of getting a quiver from double Bruhat cells in Cluster Algebras III. ( G , u , v ) � Q u , v ( call BFZ quiver ) 9 / 24
Example (BFZ quiver) Example W = S 4 , u = s 3 s 2 s 1 s 2 s 3 , v = e A 3 10 / 24
Relation to dimers Gr ( k , n ) w n ∈ S n BFZ BKM Q w n , e dimer ∼ In type A , the BFZ quivers are planar, but not true in general. 11 / 24
Quivers in other types Instead of drawing them on a plane, we will draw the BFZ quivers on the cylinders over the corresponding Dynkin digrams. n 1 2 3 n − 2 n − 1 12 / 24
Quivers in other types Instead of drawing them on a plane, we will draw the BFZ quivers on the cylinders over the corresponding Dynkin digrams. n 1 2 3 n − 2 n − 1 Theorem (K) For any symmetric Kac–Moody group G, the quiver Q u , v is planar in each sheet. 13 / 24
Example of a dimer model on a cylinder over E 7 3 7 6 5 4 u = s 1 s 3 s 2 s 4 s 5 s 7 s 3 s 6 s 1 s 5 s 7 s 6 s 4 s 3 s 2 s 1 s 4 s 5 s 6 s 7 2 1 7 6 5 4 4 2 1 4 3 14 / 24
Quivers in other types Theorem (K) • Each face of Q u , v is oriented. • Each face of Q u , v on the cylinder projects onto an edge of the Dynkin diagram. • Each edge of Q u , v projects onto a vertex of the Dynkin diagram or an edge of the Dynkin diagram. 15 / 24
The quivers Q u , v To get the quiver Q u , v , we attach the quiver Q e , v on top of the quiver Q u , e . We will see this with u = s 1 s 2 s 1 s 3 , v = s 2 s 3 s 3 s 1 ∈ S 4 . Q e , v : Q u , v : 1 2 3 Q u , e : 1 2 3 1 2 3
The quivers Q u , v To get the quiver Q u , v , we attach the quiver Q e , v on top of the quiver Q u , e . We will see this with u = s 1 s 2 s 1 s 3 , v = s 2 s 3 s 3 s 1 ∈ S 4 . Q e , v : Q u , v : 1 2 3 Q u , e : 1 2 3 1 2 3 17 / 24
The quivers Q u , v p 2 e p 2 ℓ k k F ℓ k ℓ p 1 e p 1 Figure 1: Case 1 p 2 e p 2 ℓ k k ℓ e p 1 k ℓ e p 1 Figure 2: Case 2 18 / 24
Rigid potential We need the superpotential of these quivers to be Rigid. Rigid: None of the mutations of the potential creates a 2-cycle. 19 / 24
Rigid potential We need the superpotential of these quivers to be Rigid. Rigid: None of the mutations of the potential creates a 2-cycle. Definition A potential S is called rigid if every oriented cycle in Q belongs to the Jacobian ideal J ( S ) up to cyclic equivalence. 20 / 24
Rigid potential Example (Non-example) a 1 2 S 1 = abc c d b e 3 4 J ( S 1 ) = � bc , ca , ab � . So abc ∈ J ( S 1 ) but cde / ∈ J ( S 1 ). Therefore S 1 is not rigid. 21 / 24
Rigid potential Example (Non-example) a 1 2 S 1 = abc c d b e 3 4 J ( S 1 ) = � bc , ca , ab � . So abc ∈ J ( S 1 ) but cde / ∈ J ( S 1 ). Therefore S 1 is not rigid. Example a 1 2 S 2 = abc + cde c d b e 3 4 J ( S 2 ) = � bc , ca , ab + de , ec , cd � . So abc ∈ J ( S 2 ) but cde ∈ J ( S 2 ). Therefore S 2 is rigid. 22 / 24
n 1 2 3 n − 2 n − 1 Theorem (Buan-Iyama-Reiten-Smith, K) Let g be a simply laced, star shaped Kac-Moody Lie algebra and Q u , e be the quiver corresponding to the double Bruhat decomposition. Then the superpotential of Q u , e is rigid. 23 / 24
Thank you! 24 / 24
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