Dimer models and the special McKay correspondence

Ishii, Akira; Ueda, Kanji

doi:10.2140/gt.2015.19.3405

Cited by 40 publications

(58 citation statements)

References 29 publications

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“…Proposition 3.6 (see e.g., [IU2,Section 9], [Boc3,Corollary 2.9]). There exists a one to one correspondence between the set of slopes of zigzag paths on a consistent dimer model and the set of primitive side segments of the perfect matching polygon.…”

Section: Figure 3 Extremal Perfect Matchingsmentioning

confidence: 99%

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Semi-steady non-commutative crepant resolutions via regular dimer models

Nakajima¹

2019

Algebraic Combinatorics

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A consistent dimer model gives a non-commutative crepant resolution (= NCCR) of a 3dimensional Gorenstein toric singularity. In particular, it is known that a consistent dimer model gives a class of NCCRs called steady if and only if it is homotopy equivalent to a regular hexagonal dimer model. Inspired by this result, we detect another nice property on NCCRs that characterizes square dimer models. We call such NCCRs semi-steady NCCRs, and study their properties.

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Section: Figure 3 Extremal Perfect Matchingsmentioning

confidence: 99%

“…Theorem 3.9 (see e.g., [Bro,IU2,Boc2]). Suppose that (Q, W Q ) is the QP associated with a consistent dimer model Γ and P(Q, W Q ) is the complete Jacobian algebra.…”

Section: Figure 3 Extremal Perfect Matchingsmentioning

confidence: 99%

Semi-steady non-commutative crepant resolutions via regular dimer models

Nakajima¹

2019

Algebraic Combinatorics

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“…This is distinct from the construction of Ishii-Ueda [20] for dimer models, which is a process linking the removal of a corner in a lattice polygon to the universal localisation of certain arrows in a quiver in order to determine an open subset in an associated moduli space.…”

Section: The Cornering Category and Recollementmentioning

confidence: 99%

“…Ishii-Ueda [20] and Broomhead [9] show that R admits a noncommutative crepant resolution A = End R ( i∈Q 0 M i ) obtained as the Jacobian algebra kQ/I of a quiver with potential arising from a consistent dimer model on a real two-torus. Toric algebras of this form necessarily satisfy Assumption 5.1; in fact, the conclusions of Lemma A.1 were noted first by Ishii-Ueda [22,Proposition 8.3] and the dual bundle T ∨ ∼ = i∈Q 0 T ∨ i determines a derived equivalence…”

Section: The Toric Case In Dimension Threementioning

confidence: 99%

Multigraded linear series and recollement

2017

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Given a scheme Y equipped with a collection of globally generated vector bundles E 1 , . . . , E n , we study the universal morphism from Y to a fine moduli space M(E) of cyclic modules over the endomorphism algebra of E :This generalises the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category. When the morphism is surjective, this gives a fine moduli space interpretation of the image, and as an application we show that for a small, finite subgroup G ⊂ GL(2, k), every sub-minimal partial resolution of A 2 k /G is isomorphic to a fine moduli space M(E C ) where E C is a summand of the bundle E defining the reconstruction algebra. We also consider applications to Gorenstein affine threefolds, where Reid's recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution.B Alastair Craw a.craw@bath.ac.uk

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“…It is proved in [Bro12,Dav11,Boc11] that the Jacobian algebra B = Jac(Q, W ) is a non-commutative crepant resolution of its center C = Z(B) which is the coordinate ring of a Gorenstein affine toric threefold. Moreover the coordinate ring of any Gorenstein affine toric threefold can be obtained from a consistent dimer model [Gul08,IU09].…”

Section: Examplementioning

confidence: 99%

Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday

Amiot¹,

Iyama²,

Reiten³

2015

ajm

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Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday.Abstract. By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is triangle equivalent to the 1-cluster category of the path algebra of a Dynkin quiver (i.e., the orbit category of the derived category by the action of the Auslander-Reiten translation). In this paper we give a systematic method to construct a similar type of triangle equivalence between the stable category of Cohen-Macaulay modules over a Gorenstein isolated singularity R and the generalized (higher) cluster category of a finite dimensional algebra Λ. The key role is played by a bimodule Calabi-Yau algebra, which is the higher Auslander algebra of R as well as the higher preprojective algebra of an extension of Λ. As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay R-modules and the derived category of Λ. Our main results apply in particular to a class of cyclic quotient singularities and to certain toric affine threefolds associated with dimer models.

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Dimer models and the special McKay correspondence

Cited by 40 publications

References 29 publications

Semi-steady non-commutative crepant resolutions via regular dimer models

Semi-steady non-commutative crepant resolutions via regular dimer models

Multigraded linear series and recollement

Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday

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