Chip firing on Dynkin diagrams and McKay quivers

Georgia Benkart, Caroline Klivans, Victor Reiner

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

This paper establishes new connections between the representation theory of finite groups and sandpile dynamics. Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay–Cartan matrices for finite subgroups G of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights. In the McKay–Cartan case for finite subgroups of the special linear group, the cokernel is related to the abelianization of the subgroup G. In the special case of the classical McKay correspondence, the critical group and the abelianization are shown to be isomorphic.

Original languageEnglish (US)
Pages (from-to)615-648
Number of pages34
JournalMathematische Zeitschrift
Volume290
Issue number1-2
DOIs
StatePublished - Oct 1 2018

Bibliographical note

Funding Information:
This work was inspired by group discussions at the workshop Algebraic Combinatorixx at the Banff International Research Station (BIRS) in 2011 and began at the workshop Whittaker Functions, Schubert Calculus and Crystals at the Institute for Computational and Experimental Research in Mathematics (ICERM) in 2013. The authors express their appreciation to those institutes for their support and hospitality. The second and third authors thank the CMO-BIRS-Oaxaca Institute for its hospitality during their 2015 workshop Sandpile Groups. In particular, they extend thanks to Sam Payne for a helpful discussion there leading to the formulation of Proposition?5.22 , and to Shaked Koplewitz for allowing them to sketch his proof of Proposition?6.19. The authors thank Christian Gaetz for the permission to cite results from his University of Minnesota honors baccalaureate thesis, and they thank an anonymous referee for helpful edits. The third author also thanks T. Schedler for helpful comments and gratefully acknowledges partial support by NSF Grant DMS-1001933.

Funding Information:
Acknowledgements This work was inspired by group discussions at the workshop Algebraic Combinatorixx at the Banff International Research Station (BIRS) in 2011 and began at the workshop Whittaker Functions, Schubert Calculus and Crystals at the Institute for Computational and Experimental Research in Mathematics (ICERM) in 2013. The authors express their appreciation to those institutes for their support and hospitality. The second and third authors thank the CMO-BIRS-Oaxaca Institute for its hospitality during their 2015 workshop Sandpile Groups. In particular, they extend thanks to Sam Payne for a helpful discussion there leading to the formulation of Proposition 5.22, and to Shaked Koplewitz for allowing them to sketch his proof of Proposition 6.19. The authors thank Christian Gaetz for the permission to cite results from his University of Minnesota honors baccalaureate thesis, and they thank an anonymous referee for helpful edits. The third author also thanks T. Schedler for helpful comments and gratefully acknowledges partial support by NSF Grant DMS-1001933.

Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

Keywords

  • Abelianization
  • Avalanche-finite matrix
  • Chip firing
  • Dynkin diagram
  • Highest root
  • M-matrix
  • McKay correspondence
  • McKay quiver
  • Minuscule weight
  • Numbers game
  • Root system
  • Sandpile
  • Toppling
  • Z-matrix

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