tailieunhanh - Báo cáo toán học: "A Relative Laplacian spectral recursion"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A Relative Laplacian spectral recursion. | A Relative Laplacian spectral recursion Art M. Duval Department of Mathematical Sciences University of Texas at El Paso El Paso TX 79968-0514 artduval@ Submitted Jun 30 2005 Accepted Jan 13 2006 Published Feb 1 2006 Mathematics Subject Classification Primary 15A18 Secondary 55U10 06A07 05E99. Keywords Laplacian spectra matroid complex shifted simplicial complex relative simplicial pair. Dedicated to Richard Stanley on the occasion of his 60th birthday. Abstract The Laplacian spectral recursion satisfied by matroid complexes and shifted complexes expresses the eigenvalues of the combinatorial Laplacian of a simplicial complex in terms of its deletion and contraction with respect to vertex e and the relative simplicial pair of the deletion modulo the contraction. We generalize this recursion to relative simplicial pairs which we interpret as convex subsets of the Boolean algebra. The deletion modulo contraction term is replaced by the result of removing from the convex set T all pairs of faces in T that differ only by vertex e. We show that shifted pairs and some matroid pairs satisfy this recursion. We also show that the class of convex sets satisfying this recursion is closed under a wide variety of operations including duality and taking skeleta. 1 Introduction There are two good reasons to extend the Laplacian spectral recursion from simplicial complexes to relative simplicial pairs. The spectral recursion for simplicial complexes expresses the eigenvalues of the combinatorial Laplacian dd d d of a simplicial complex A in terms of the eigenvalues of its deletion A e contraction A e and an error term A e A e . This recursion does not hold for all simplicial complexes but does hold for independence complexes of matroids and shifted simplicial complexes 2 . In each case the deletion and contraction are again matroids or shifted complexes respectively but the error term is only a relative simplicial pair of the appropriate kind of complexes. Being able .