Homology character of the parabolic coset poset
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912587694735360 |
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| author | Douvropoulos, Theo Josuat-Vergès, Matthieu |
| author_facet | Douvropoulos, Theo Josuat-Vergès, Matthieu |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_11905 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homology character of the parabolic coset poset Douvropoulos, Theo Josuat-Vergès, Matthieu Combinatorics 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. |
| title | Homology character of the parabolic coset poset |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.11905 |