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 ORCID3

  • 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.


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]

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