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## 2022 - Francophone Computer Algebra Days - Journées nationales de calcul formel

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## Combinatorial Reid's recipe for consistent dimer models

Appears in collection : Algebraic geometry and complex geometry / Géométrie algébrique et géométrie complexe

Reid's recipe is an equivalent of the McKay correspondence in dimension three. It marks interior line segments and lattice points in the fan of the G-Hilbert scheme (a specific crepant resolution of $\mathbb{C}^{3} / G$ for $G \subset S L(3, \mathbb{C})$ ) with characters of irreducible representations of $G$. Our goal is to generalise this by marking the toric fan of a crepant resolution of any affine Gorenstein singularity, in a way that is compatible with both the G-Hilbert case and its categorical counterpart known as Derived Reid's Recipe. To achieve this, we foray into the combinatorial land of quiver moduli spaces and dimer models. This is joint work with Alastair Craw and Jesus Tapia Amador.

### Citation data

• DOI 10.24350/CIRM.V.19983603
• Cite this video Heuberger Liana (11/29/22). Combinatorial Reid's recipe for consistent dimer models. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19983603
• URL https://dx.doi.org/10.24350/CIRM.V.19983603

### Bibliography

• AMADOR, Jesus Tapia, HEUBERGER, Liana, et CRAW, Alastair. Combinatorial Reid's recipe for consistent dimer models. Épijournal de Géométrie Algébrique, 2021, vol. 5. - https://doi.org/10.46298/epiga.2021.volume5.6085
• ISHII, A. et UEDA, K. On moduli spaces of quiver representations associated with brane tilings Higher dimensional algebraic varieties and vector bundles, 127141, RIMS Kokyuroku Bessatsu, B9. Res. Inst. Math. Sci.(RIMS), Kyoto, 2008. - http://hdl.handle.net/2433/176749
• CRAW, Alastair. An explicit construction of the McKay correspondence for A-Hilb C3. Journal of Algebra, 2005, vol. 285, no 2, p. 682-705. - https://doi.org/10.1016/j.jalgebra.2004.10.001
• BOCKLANDT, Raf, CRAW, Alastair, et QUINTERO VÉLEZ, Alexander. Geometric Reid's recipe for dimer models. Mathematische Annalen, 2015, vol. 361, no 3, p. 689-723. - http://dx.doi.org/10.1007/s00208-014-1085-8
• BOCKLANDT, Raf, CRAW, Alastair, et QUINTERO VÉLEZ, Alexander. Correction to: Geometric Reid's recipe for dimer models. Mathematische Annalen, 2021, vol. 380, no 1, p. 911-913. - http://dx.doi.org/10.1007/s00208-020-02127-w
• CAUTIS, Sabin, CRAW, Alastair, et LOGVINENKO, Timothy. Derived Reid's recipe for abelian subgroups of SL3 (ℂ). Journal für die reine und angewandte Mathematik (Crelles Journal), 2017, vol. 2017, no 727, p. 1-48. - https://doi.org/10.1515/crelle-2014-0086

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