2025/09/15 by Theo Douvropoulos, Douvropoulos, Theo, Matthieu Josuat-Vergès +1
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2509.11905
openalex publication_date 2025/09/15 · openalex created_date 2025/10/12 · openalex updated_date 2026/08/05
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 Möbius 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.