Combinatorial Laplacians of matroid complexes

Authors:
W. Kook, V. Reiner and D. Stanton

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

MSC (2000):
Primary 05B35

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

Published electronically:
September 13, 1999

MathSciNet review:
1697094

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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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Additional Information

**W. Kook**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Address at time of publication:
Department of Mathematics, The George Washington University, Washington DC 20052

Email:
kook@math.umn.edu, andrewk@gwu.edu

**V. Reiner**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
reiner@math.umn.edu

**D. Stanton**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
stanton@math.umn.edu

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

Keywords:
Matroid,
matroid complex,
Laplacian,
internal activity,
external activity

Received by editor(s):
July 16, 1997

Received by editor(s) in revised form:
June 30, 1999

Published electronically:
September 13, 1999

Additional Notes:
The second author was supported by Sloan Foundation and University of Minnesota McKnight Land Grant Fellowships. The third author was supported by NSF grant DMS-9400510.

Article copyright:
© Copyright 1999
American Mathematical Society