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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.04021 |
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| _version_ | 1866917829664571392 |
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| author | Hoyer, Linda |
| author_facet | Hoyer, Linda |
| contents | For every partition $λ$ of a positive integer $n$, let $S^λ$ be the corresponding Specht module of the symmetric group $\mathfrak{S}_n$, and let $\det(λ)\in \mathbb Z$ denote the Gram determinant of the canonical bilinear form with respect to the standard basis of $S^λ$. Writing $\det(λ)=m \cdot 2^{a_λ^{(2)}}$ for integers $a_λ^{(2)}$ and $m$ with $m$ odd, we show that if the dimension of $S^λ$ is even, then $a_λ^{(2)}$ is also even. This confirms a conjecture posed by Richard Parker in the special case of the symmetric groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04021 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Gram determinants of the Specht modules Hoyer, Linda Combinatorics Representation Theory 05E10 For every partition $λ$ of a positive integer $n$, let $S^λ$ be the corresponding Specht module of the symmetric group $\mathfrak{S}_n$, and let $\det(λ)\in \mathbb Z$ denote the Gram determinant of the canonical bilinear form with respect to the standard basis of $S^λ$. Writing $\det(λ)=m \cdot 2^{a_λ^{(2)}}$ for integers $a_λ^{(2)}$ and $m$ with $m$ odd, we show that if the dimension of $S^λ$ is even, then $a_λ^{(2)}$ is also even. This confirms a conjecture posed by Richard Parker in the special case of the symmetric groups. |
| title | On the Gram determinants of the Specht modules |
| topic | Combinatorics Representation Theory 05E10 |
| url | https://arxiv.org/abs/2411.04021 |