Homology character of the parabolic coset poset

Abstract: Motivated by the analogy with the Coxeter complex on one side, and parking functions on the other side, we study the poset of parabolic cosets in a finite Coxeter group. We show that this poset is Cohen-Macaulay, and get an explicit formula for the character of its (unique) nonzero homology group in terms of the M\"obius function of the intersection lattice. This homology character becomes a positive element of the parabolic Burnside ring (in its natural basis) after tensoring with the sign character. The coefficients of this character essentially encode the colored $h$-vector of the positive chamber complex (following Bastidas, Hohlweg, and Saliola, this complex is defined by taking Weyl chambers that lie on the positive side of a generic hyperplane). Roughly speaking, tensoring by the sign character on one side corresponds to the transformation going from the $f$-vector to the $h$-vector on the other side.