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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.20673 |
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| _version_ | 1866913938655936512 |
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| author | Wang, Frank Yee, Eric |
| author_facet | Wang, Frank Yee, Eric |
| contents | We compute the Hilbert series of the space of $n=3$ variable quasi-invariant polynomials in characteristic $2$ and $3$, capturing the dimension of the homogeneous components of the space, and explicitly describe the generators in the characteristic $2$ case. In doing so we extend the work of the first author in 2023 on quasi-invariant polynomials in characteristic $p>n$ and prove that a sufficient condition found by Ren-Xu in 2020 on when the Hilbert series differs between characteristic $0$ and $p$ is also necessary for $n=3$, $p=2,3$. This is the first description of quasi-invariant polynomials in the case, where the space forms a modular representation over the symmetric group, bringing us closer to describing the quasi-invariant polynomials in all characteristics and numbers of variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20673 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hilbert Series of $S_3$-Quasi-Invariant Polynomials in Characteristics 2, 3 Wang, Frank Yee, Eric Representation Theory Quantum Algebra 16S38 We compute the Hilbert series of the space of $n=3$ variable quasi-invariant polynomials in characteristic $2$ and $3$, capturing the dimension of the homogeneous components of the space, and explicitly describe the generators in the characteristic $2$ case. In doing so we extend the work of the first author in 2023 on quasi-invariant polynomials in characteristic $p>n$ and prove that a sufficient condition found by Ren-Xu in 2020 on when the Hilbert series differs between characteristic $0$ and $p$ is also necessary for $n=3$, $p=2,3$. This is the first description of quasi-invariant polynomials in the case, where the space forms a modular representation over the symmetric group, bringing us closer to describing the quasi-invariant polynomials in all characteristics and numbers of variables. |
| title | Hilbert Series of $S_3$-Quasi-Invariant Polynomials in Characteristics 2, 3 |
| topic | Representation Theory Quantum Algebra 16S38 |
| url | https://arxiv.org/abs/2412.20673 |