Hyperoctahedral Eulerian Idempotents, Hodge Decompositions, and Signed Graph Coloring Complexes

Phil Hanlon proved that the coefficients of the chromatic polynomial of a graph G are equal (up to sign) to the dimensions of the summands in a Hodge-type decomposition of the top homology of the coloring complex for G. We prove a type B analogue of this result for chromatic polynomials of signed graphs using hyperoctahedral Eulerian idempotents

Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices

We prove that every infinite sequence of skew-symmetric or symmetric matrices M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such that M_i is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in M_j, if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymour's theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle's theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum's theorem for graphs of bounded rank-width with respect to pivot-minors.Comment: 43 page

A graph theoretic approach to graded identities for matrices

We consider the algebra M_k(C) of k-by-k matrices over the complex numbers and view it as a crossed product with a group G of order k by embedding G in the symmetric group S_k via the regular representation and embedding S_k in M_k(C) in the usual way. This induces a natural G-grading on M_k(C) which we call a crossed product grading. This grading is the so called elementary grading defined by any k-tuple (g_1,g_2,..., g_k) of distinct elements g_i in G. We study the graded polynomial identities for M_k(C) equipped with a crossed product grading. To each multilinear monomial in the free graded algebra we associate a directed labeled graph. This approach allows us to give new proofs of known results of Bahturin and Drensky on the generators of the T-ideal of identities and the Amitsur-Levitsky Theorem. Our most substantial new result is the determination of the asymptotic formula for the G-graded codimension of M_k(C).Comment: 21 pages, 3 figure