Combinatorial Reid's recipe for consistent dimer models

Alastair Craw; Liana Heuberger; Jesus Tapia Amador

Combinatorial Reid's recipe for consistent dimer models

Alastair Craw, Liana Heuberger, Jesus Tapia Amador

Loughborough University

Research output: Contribution to journal › Article › peer-review

Abstract

Reid's recipe [Rei97, Cra05] for a finite abelian subgroup G ⊂ SL(3,C) is a combinatorial procedure that marks the toric fan of the G-Hilbert scheme with irreducible representations of G. The geometric McKay correspondence conjecture of Cautis-Logvinenko [CL09] that describes certain objects in the derived category of G-Hilb in terms of Reid's recipe was later proved by Logvinenko et. al. [Log10, CCL17]. We generalise Reid's recipe to any consistent dimer model by marking the toric fan of a crepant resolution of the vaccuum moduli space in a manner that is compatible with the geometric correspondence of Bocklandt-Craw-Quintero-Vélez [BCQ15]. Our main tool generalises the jigsaw transformations of Nakamura [Nak01] to consistent dimer models.

Original language	French
Article number	4
Number of pages	29
Journal	Épijournal de Géométrie Algébrique
Volume	5
Early online date	26 Feb 2021
Publication status	Published - 26 Feb 2021

Bibliographical note

Funding Information:
The second author was funded by grant EP/S004130/1 (PI Elisa Postinghel) and the third author was funded by CONACYT.

Publisher Copyright:
© by the author(s) This work is licensed under http://creativecommons.org/licenses/by-sa/4.0/

Keywords

Dimer model
Jigsaw transformations
Quiver moduli space
Reid's recipe
Tilting bundle

ASJC Scopus subject areas

Geometry and Topology
Algebra and Number Theory

Access to Document

https://arxiv.org/pdf/2001.07506.pdfLicence: Unspecified
https://epiga.episciences.org/7224Licence: CC BY-SA

Cite this

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title = "Combinatorial Reid's recipe for consistent dimer models",

abstract = "Reid's recipe [Rei97, Cra05] for a finite abelian subgroup G ⊂ SL(3,C) is a combinatorial procedure that marks the toric fan of the G-Hilbert scheme with irreducible representations of G. The geometric McKay correspondence conjecture of Cautis-Logvinenko [CL09] that describes certain objects in the derived category of G-Hilb in terms of Reid's recipe was later proved by Logvinenko et. al. [Log10, CCL17]. We generalise Reid's recipe to any consistent dimer model by marking the toric fan of a crepant resolution of the vaccuum moduli space in a manner that is compatible with the geometric correspondence of Bocklandt-Craw-Quintero-V{\'e}lez [BCQ15]. Our main tool generalises the jigsaw transformations of Nakamura [Nak01] to consistent dimer models.",

keywords = "Dimer model, Jigsaw transformations, Quiver moduli space, Reid's recipe, Tilting bundle",

author = "Alastair Craw and Liana Heuberger and {Tapia Amador}, Jesus",

note = "Funding Information: The second author was funded by grant EP/S004130/1 (PI Elisa Postinghel) and the third author was funded by CONACYT. Publisher Copyright: {\textcopyright} by the author(s) This work is licensed under http://creativecommons.org/licenses/by-sa/4.0/",

year = "2021",

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day = "26",

language = "French",

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journal = "{\'E}pijournal de G{\'e}om{\'e}trie Alg{\'e}brique",

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AU - Craw, Alastair

AU - Heuberger, Liana

AU - Tapia Amador, Jesus

N1 - Funding Information: The second author was funded by grant EP/S004130/1 (PI Elisa Postinghel) and the third author was funded by CONACYT. Publisher Copyright: © by the author(s) This work is licensed under http://creativecommons.org/licenses/by-sa/4.0/

PY - 2021/2/26

Y1 - 2021/2/26

N2 - Reid's recipe [Rei97, Cra05] for a finite abelian subgroup G ⊂ SL(3,C) is a combinatorial procedure that marks the toric fan of the G-Hilbert scheme with irreducible representations of G. The geometric McKay correspondence conjecture of Cautis-Logvinenko [CL09] that describes certain objects in the derived category of G-Hilb in terms of Reid's recipe was later proved by Logvinenko et. al. [Log10, CCL17]. We generalise Reid's recipe to any consistent dimer model by marking the toric fan of a crepant resolution of the vaccuum moduli space in a manner that is compatible with the geometric correspondence of Bocklandt-Craw-Quintero-Vélez [BCQ15]. Our main tool generalises the jigsaw transformations of Nakamura [Nak01] to consistent dimer models.

AB - Reid's recipe [Rei97, Cra05] for a finite abelian subgroup G ⊂ SL(3,C) is a combinatorial procedure that marks the toric fan of the G-Hilbert scheme with irreducible representations of G. The geometric McKay correspondence conjecture of Cautis-Logvinenko [CL09] that describes certain objects in the derived category of G-Hilb in terms of Reid's recipe was later proved by Logvinenko et. al. [Log10, CCL17]. We generalise Reid's recipe to any consistent dimer model by marking the toric fan of a crepant resolution of the vaccuum moduli space in a manner that is compatible with the geometric correspondence of Bocklandt-Craw-Quintero-Vélez [BCQ15]. Our main tool generalises the jigsaw transformations of Nakamura [Nak01] to consistent dimer models.

KW - Dimer model

KW - Jigsaw transformations

KW - Quiver moduli space

KW - Reid's recipe

KW - Tilting bundle

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Combinatorial Reid's recipe for consistent dimer models

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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Cite this