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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
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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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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

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