Critical Groups of Simplicial Complexes

Art M. Duval; Caroline J. Klivans; Jeremy L. Martin

doi:10.46298/dmtcs.2909

Art M. Duval ; Caroline J. Klivans ; Jeremy L. Martin - Critical Groups of Simplicial Complexes

dmtcs:2909 - Discrete Mathematics & Theoretical Computer Science, January 1, 2011, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) - https://doi.org/10.46298/dmtcs.2909

Critical Groups of Simplicial ComplexesArticle

Authors: Art M. Duval ¹; Caroline J. Klivans ²; Jeremy L. Martin ³

1 Department of Mathematical Sciences
2 Department of Computer Science, University of Chicago
3 Department of Mathematics

We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian operators, generalizing the construction of the critical group of a graph. We show how to realize these critical groups explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe how the critical groups of a complex represent flow along its faces, and sketch another potential interpretation as analogues of Chow groups.

https://doi.org/10.46298/dmtcs.2909

Source: HAL:hal-01215108v1

Volume: DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)

Section: Proceedings

Published on: January 1, 2011

Imported on: January 31, 2017

Keywords: graph,simplicial complex,critical group,combinatorial Laplacian,chip-firing game,sandpile model,spanning trees,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

Bibliographic References

18 Documents citing this article

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