A charge monomial basis of the Garsia-Procesi ring
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917057360035840 |
|---|---|
| author | Hanada, Mitsuki |
| author_facet | Hanada, Mitsuki |
| contents | We construct a basis of the Garsia-Procesi ring using the catabolizability type of standard Young tableaux and the charge statistic. This basis turns out to be equal to the descent basis defined in Carlsson-Chou (2024+). Our new construction connects the combinatorics of the basis with the well-known combinatorial formula for the modified Hall-Littlewood polynomials $\tilde{H}_μ[X;q]$, due to Lascoux, which expresses the polynomials as a sum over standard tableaux that satisfy a catabolizability condition. In addition, we prove that identifying a basis for the antisymmetric part of $R_μ$ with respect to a Young subgroup $S_γ$ is equivalent to finding pairs of standard tableaux that satisfy conditions regarding catabolizability and descents. This gives an elementary proof of the fact that the graded Frobenius character of $R_μ$ is given by the catabolizability formula for $\tilde{H}_μ[X;q]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15514 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A charge monomial basis of the Garsia-Procesi ring Hanada, Mitsuki Combinatorics 05E10 (Primary) 05E05, 05A30 (Secondary) We construct a basis of the Garsia-Procesi ring using the catabolizability type of standard Young tableaux and the charge statistic. This basis turns out to be equal to the descent basis defined in Carlsson-Chou (2024+). Our new construction connects the combinatorics of the basis with the well-known combinatorial formula for the modified Hall-Littlewood polynomials $\tilde{H}_μ[X;q]$, due to Lascoux, which expresses the polynomials as a sum over standard tableaux that satisfy a catabolizability condition. In addition, we prove that identifying a basis for the antisymmetric part of $R_μ$ with respect to a Young subgroup $S_γ$ is equivalent to finding pairs of standard tableaux that satisfy conditions regarding catabolizability and descents. This gives an elementary proof of the fact that the graded Frobenius character of $R_μ$ is given by the catabolizability formula for $\tilde{H}_μ[X;q]$. |
| title | A charge monomial basis of the Garsia-Procesi ring |
| topic | Combinatorics 05E10 (Primary) 05E05, 05A30 (Secondary) |
| url | https://arxiv.org/abs/2410.15514 |