Higher Specht bases for generalizations of the coinvariant ring
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866916115940114432 |
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| author | Gillespie, Maria Rhoades, Brendon |
| author_facet | Gillespie, Maria Rhoades, Brendon |
| contents | The classical coinvariant ring $R_n$ is defined as the quotient of a polynomial ring in $n$ variables by the positive-degree $S_n$-invariants. It has a known basis that respects the decomposition of $R_n$ into irreducible $S_n$-modules, consisting of the higher specht polynomials due to Ariki, Terasoma, and Yamada.
We provide an extension of the higher Specht basis to the generalized coinvariant rings $R_{n,k}$. We also give a conjectured higher Specht basis for the Garsia-Procesi modules $R_μ$, and provide a proof of the conjecture in the case of two-row partition shapes $μ$. We then combine these results to give a higher Specht basis for an infinite subfamily of the modules $R_{n,k,μ}$ recently defined by Griffin, which are a common generalization of $R_{n,k}$ and $R_μ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_02110 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Higher Specht bases for generalizations of the coinvariant ring Gillespie, Maria Rhoades, Brendon Combinatorics 05E10 (Primary) 05E05, 05E40, 20C30 (Secondary) The classical coinvariant ring $R_n$ is defined as the quotient of a polynomial ring in $n$ variables by the positive-degree $S_n$-invariants. It has a known basis that respects the decomposition of $R_n$ into irreducible $S_n$-modules, consisting of the higher specht polynomials due to Ariki, Terasoma, and Yamada. We provide an extension of the higher Specht basis to the generalized coinvariant rings $R_{n,k}$. We also give a conjectured higher Specht basis for the Garsia-Procesi modules $R_μ$, and provide a proof of the conjecture in the case of two-row partition shapes $μ$. We then combine these results to give a higher Specht basis for an infinite subfamily of the modules $R_{n,k,μ}$ recently defined by Griffin, which are a common generalization of $R_{n,k}$ and $R_μ$. |
| title | Higher Specht bases for generalizations of the coinvariant ring |
| topic | Combinatorics 05E10 (Primary) 05E05, 05E40, 20C30 (Secondary) |
| url | https://arxiv.org/abs/2005.02110 |