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.