Chow rings and augmented Chow rings of uniform matroids and their $q$-analogs
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arXiv
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| Format: | Preprint |
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2024
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| author | Liao, Hsin-Chieh |
| author_facet | Liao, Hsin-Chieh |
| contents | We study the Hilbert series and the representations of $\mathfrak{S}_n$ and $GL_n(\mathbb{F}_q)$ on the (augmented) Chow rings of uniform matroids $U_{r,n}$ and $q$-uniform matroids $U_{r,n}(q)$. The Frobenius series for uniform matroids and their $q$-analogs are computed. As a byproduct, we recover Hameister, Rao, and Simpson's formula for the Hilbert series of Chow rings of $q$-uniform matroids in terms of permutations and further obtain their augmented counterpart in terms of decorated permutations.
We also show that the equivariant Charney--Davis quantity of the (augmented) Chow ring of a matroid is nonnegative (i.e., a genuine representation of a group of automorphisms of the matroid). When the matroid is a uniform matroid and the group is $\mathfrak{S}_n$, the representation either vanishes or is a Foulkes representation (i.e., a Specht module of a ribbon shape). Specializing to the usual Charney--Davis quantities, we obtain an elegant combinatorial interpretation of Hameister, Rao, and Simpson's formula for Chow rings of $q$-uniform matroids and its augmented counterpart. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19660 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chow rings and augmented Chow rings of uniform matroids and their $q$-analogs Liao, Hsin-Chieh Combinatorics 05B35, 05E18, 05E05, 05E14 We study the Hilbert series and the representations of $\mathfrak{S}_n$ and $GL_n(\mathbb{F}_q)$ on the (augmented) Chow rings of uniform matroids $U_{r,n}$ and $q$-uniform matroids $U_{r,n}(q)$. The Frobenius series for uniform matroids and their $q$-analogs are computed. As a byproduct, we recover Hameister, Rao, and Simpson's formula for the Hilbert series of Chow rings of $q$-uniform matroids in terms of permutations and further obtain their augmented counterpart in terms of decorated permutations. We also show that the equivariant Charney--Davis quantity of the (augmented) Chow ring of a matroid is nonnegative (i.e., a genuine representation of a group of automorphisms of the matroid). When the matroid is a uniform matroid and the group is $\mathfrak{S}_n$, the representation either vanishes or is a Foulkes representation (i.e., a Specht module of a ribbon shape). Specializing to the usual Charney--Davis quantities, we obtain an elegant combinatorial interpretation of Hameister, Rao, and Simpson's formula for Chow rings of $q$-uniform matroids and its augmented counterpart. |
| title | Chow rings and augmented Chow rings of uniform matroids and their $q$-analogs |
| topic | Combinatorics 05B35, 05E18, 05E05, 05E14 |
| url | https://arxiv.org/abs/2406.19660 |