The Hodge structure of the coloring complex of a hypergraph

Let G be a simple graph with n vertices. The coloring complex Δ(G) was defined by Steingrímsson, and the homology of Δ(G) was shown to be nonzero only in dimension n−3 by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group H_n−3(Δ(G)) where the dimension of the jth component in the decomposition, , equals the absolute value of the coefficient of λ^j in the chromatic polynomial of G, χ_G(λ).Let H be a hypergraph with n vertices. In this paper, we define the coloring complex of a hypergraph, Δ(H), and show that the coefficient of λ^j in χ_H(λ) gives the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of Δ(H). We also examine conditions on a hypergraph, H, for which its Hodge subcomplexes are Cohen–Macaulay, and thus where the absolute value of the coefficient of λ^j in χ_H(λ) equals the dimension of the jth Hodge piece of the Hodge decomposition of Δ(H). We also note that the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of the intersection of coloring complexes is given by the coefficient of jth term in the associated chromatic polynomial.