Higher Specht polynomials under the diagonal action
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909098811850752 |
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| author | Gillespie, Maria |
| author_facet | Gillespie, Maria |
| contents | We introduce higher Specht polynomials - analogs of Specht polynomials in higher degrees - in two sets of variables $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$ under the diagonal action of the symmetric group $S_n$. This generalizes the classical Specht polynomial construction in one set of variables, as well as the higher Specht basis for the coinvariant ring $R_n$ due to Ariki, Terasoma, and Yamada, which has the advantage of respecting the decomposition into irreducibles.
As our main application of the general theory, we provide a higher Specht basis for the hook shape Garsia--Haiman modules. In the process, we obtain a new formula for their doubly graded Frobenius series in terms of new generalized cocharge statistics on tableaux. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher Specht polynomials under the diagonal action Gillespie, Maria Combinatorics 05E10 (Primary) 05E05, 05E40, 20C30 (Secondary) We introduce higher Specht polynomials - analogs of Specht polynomials in higher degrees - in two sets of variables $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$ under the diagonal action of the symmetric group $S_n$. This generalizes the classical Specht polynomial construction in one set of variables, as well as the higher Specht basis for the coinvariant ring $R_n$ due to Ariki, Terasoma, and Yamada, which has the advantage of respecting the decomposition into irreducibles. As our main application of the general theory, we provide a higher Specht basis for the hook shape Garsia--Haiman modules. In the process, we obtain a new formula for their doubly graded Frobenius series in terms of new generalized cocharge statistics on tableaux. |
| title | Higher Specht polynomials under the diagonal action |
| topic | Combinatorics 05E10 (Primary) 05E05, 05E40, 20C30 (Secondary) |
| url | https://arxiv.org/abs/2402.05221 |