Hyperoctahedral Eulerian idempotents, Hodge decompositions, and signed graph coloring complexes (Q405237)

Summary: \textit{P. Hanlon} [Proc. Am. Math. Soc. 136, No. 11, 3741--3749 (2008; Zbl 1146.05023)] 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.