Combinatorial Laplacians of matroid complexes
Tóm tắt
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.
Từ khóa
Tài liệu tham khảo
Björner, Anders, 1992, The homology and shellability of matroids and geometric lattices, 226, 10.1017/CBO9780511662041.008
T. Brylawski, Constructions, in Theory of Matroids (ed. N. White), Encyclopedia of Mathematics and Its Applications, 26, Cambridge Univ. Press, 1986.
Brylawski, Thomas, 1992, The Tutte polynomial and its applications, 123, 10.1017/CBO9780511662041.007
Etienne, Gwihen, 1998, External and internal elements of a matroid basis, Discrete Math., 179, 111, 10.1016/S0012-365X(95)00332-Q
1996, Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing
Friedman, J., 1998, Computing Betti numbers via combinatorial Laplacians, Algorithmica, 21, 331, 10.1007/PL00009218
J. Friedman and P. Hanlon, On the Betti numbers of chessboard complexes, J. Algebraic Combin. 8 (1998), 193–203.
W. Kook, Categories of acyclic graphs and automorphisms of free groups, Ph.D. thesis (G. Carlsson, advisor), Stanford Univ., 1996.
Sagan, Bruce E., 1991, The symmetric group
Solomon, Louis, 1969, The Steinberg character of a finite group with 𝐵𝑁-pair, 213