A descent basis for the Garsia-Procesi module
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911811882713088 |
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| author | Carlsson, Erik Chou, Raymond |
| author_facet | Carlsson, Erik Chou, Raymond |
| contents | We assign to each Young diagram $λ$ a subset $\mathcal{B}_{λ'}$ of the collection of Garsia-Stanton descent monomials, and prove that it determines a basis of the Garsia-Procesi module $R_λ$, whose graded character is the Hall-Littlewood polynomial $\tilde{H}_λ[X;t]$. This basis is a major index analogue of the basis $\mathcal{B}_λ\subset R_λ$ defined by certain recursions in due to Garsia and Procesi, in the same way that the descent basis is related to the Artin basis of the coinvariant algebra $R_n$, which in fact corresponds to the case when $λ=1^n$. By anti-symmetrizing a subset of this basis with respect to the corresponding Young subgroup under the Springer action, we obtain a basis in the parabolic case, as well as a corresponding formula for the expansion of $\tilde{H}_λ[X;t]$. Despite a similar appearance, it does not appear obvious how to connect these formulas appear to the specialization of the modified Macdonald formula of Haglund, Haiman and Loehr at $q=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16278 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A descent basis for the Garsia-Procesi module Carlsson, Erik Chou, Raymond Representation Theory Combinatorics 33D52, 05E10, 05A30 We assign to each Young diagram $λ$ a subset $\mathcal{B}_{λ'}$ of the collection of Garsia-Stanton descent monomials, and prove that it determines a basis of the Garsia-Procesi module $R_λ$, whose graded character is the Hall-Littlewood polynomial $\tilde{H}_λ[X;t]$. This basis is a major index analogue of the basis $\mathcal{B}_λ\subset R_λ$ defined by certain recursions in due to Garsia and Procesi, in the same way that the descent basis is related to the Artin basis of the coinvariant algebra $R_n$, which in fact corresponds to the case when $λ=1^n$. By anti-symmetrizing a subset of this basis with respect to the corresponding Young subgroup under the Springer action, we obtain a basis in the parabolic case, as well as a corresponding formula for the expansion of $\tilde{H}_λ[X;t]$. Despite a similar appearance, it does not appear obvious how to connect these formulas appear to the specialization of the modified Macdonald formula of Haglund, Haiman and Loehr at $q=0$. |
| title | A descent basis for the Garsia-Procesi module |
| topic | Representation Theory Combinatorics 33D52, 05E10, 05A30 |
| url | https://arxiv.org/abs/2403.16278 |