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

Let C; denote a finite graph and xc(A) its chromatic polynomial. The coloring complex Delta G was defined by Einar Steingrfiusson 1181 in order to provide a Hilbert-polynomial interpretation of xo(A). While Steingrfmsson's original definition of Delta G was moHvated by algebraic considerations, the coloring complex can also be obtained as the link complex for a hyperplane arrangement, using techniques developed by Jurgen Herzog, Vic Reiner. and Volkmar Welker [11]. Coloring complexes have many interesting properties. Jakob Jonsson proved [13] that Delta G is homotopy equivalent to a wedge of spheres in fixed dimension, with the number of spheres being one less than the number of acyclic orientations of C. Axel Hultman [12] proved that Delta G, and in general any link complex for a