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

Abstract

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.