Combinatorial Reid's recipe for consistent dimer models

Abstract

Reid's recipe for a finite abelian subgroup $G\subset \text{SL}(3,\mathbb{C})$ 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 that describes certain objects in the derived category of $G\text{-Hilb}$ in terms of Reid's recipe was later proved by Logvinenko et al. 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. Our main tool generalises the jigsaw transformations of Nakamura to consistent dimer models.

29 pages, published version

References in corpus (6)

• On moduli spaces of quiver representations associated with dimer models
• Crepant resolutions and brane tilings I: Toric realization
• A note on consistency conditions on dimer models
• Crepant resolutions and brane tilings II: Tilting bundles
• Families of G-constellations over resolutions of quotient singularities
• Combinatorial Reid's recipe for consistent dimer models

Cited by in corpus (3)

• Combinatorial Reid's recipe for consistent dimer models
• Wall-crossing for iterated Hilbert schemes (or 'Hilb of Hilb')
• Gale duality and the linearisation map for noncommutative crepant resolutions