Combinatorial Laplacians of matroid complexes

Kook, W.; Reiner, V.; Stanton, D.

doi:10.1090/S0894-0347-99-00316-1

Combinatorial Laplacians of matroid complexes
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by W. Kook, V. Reiner and D. Stanton

J. Amer. Math. Soc. 13 (2000), 129-148

DOI: https://doi.org/10.1090/S0894-0347-99-00316-1

Published electronically: September 13, 1999

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Abstract:

We combinatorially interpret the spectra of discrete Laplace operators from the boundary maps in the simplicial complex of independent sets of a matroid. The interpretation follows from a surprising orthogonal decomposition of the simplicial chain groups. This decomposition is in general finer than the spectral decomposition. As a consequence, the spectra are integral. One corollary to our combinatorial interpretation may be paraphrased as stating that one can “hear" the characteristic polynomial of a matroid.

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