Equivariant gamma-positivity of matroid Chow rings
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909010714689536 |
|---|---|
| author | Liao, Hsin-Chieh |
| author_facet | Liao, Hsin-Chieh |
| contents | In this paper, we prove that the Chow ring and augmented Chow ring of a matroid are equivariantly $γ$-positive under the action of any group of automorphisms. Our approach provides an explicit combinatorial interpretation of the coefficients in the equivariant $γ$-expansion, which is new even in the non-equivariant setting. This result confirms a conjecture of Angarone, Nathanson, and Reiner, and extends the author's previous work on the positivity of equivariant Charney--Davis quantities for matroids. Specializing our formulas to uniform matroids, we obtain representation-theoretic interpretations that extend the Schur-$γ$-positivity results of Shareshian and Wachs for Eulerian and binomial Eulerian quasisymmetric functions. Finally, we address a problem posed by Athanasiadis by giving a combinatorial interpretation of a $(p,q)$-analog of the $γ$-expansion of the binomial Eulerian polynomial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivariant gamma-positivity of matroid Chow rings Liao, Hsin-Chieh Combinatorics 05B35, 05E14, 05E18, 05E05 In this paper, we prove that the Chow ring and augmented Chow ring of a matroid are equivariantly $γ$-positive under the action of any group of automorphisms. Our approach provides an explicit combinatorial interpretation of the coefficients in the equivariant $γ$-expansion, which is new even in the non-equivariant setting. This result confirms a conjecture of Angarone, Nathanson, and Reiner, and extends the author's previous work on the positivity of equivariant Charney--Davis quantities for matroids. Specializing our formulas to uniform matroids, we obtain representation-theoretic interpretations that extend the Schur-$γ$-positivity results of Shareshian and Wachs for Eulerian and binomial Eulerian quasisymmetric functions. Finally, we address a problem posed by Athanasiadis by giving a combinatorial interpretation of a $(p,q)$-analog of the $γ$-expansion of the binomial Eulerian polynomial. |
| title | Equivariant gamma-positivity of matroid Chow rings |
| topic | Combinatorics 05B35, 05E14, 05E18, 05E05 |
| url | https://arxiv.org/abs/2408.00745 |