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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2407.06603 |
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| _version_ | 1866913422907539456 |
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| author | Arrondo, Enrique Tocino, Alicia |
| author_facet | Arrondo, Enrique Tocino, Alicia |
| contents | Given a vector space $V$ over a field $\K$ whose characteristic is coprime with $d!$, let us decompose the vector space of multilinear forms $V^*\otimes\overset{\text(d)}{\ldots}\otimes V^*=\bigoplus _λW_λ(X,\K)$ according to the different partitions $λ$ of $d$, i.e. the different representations of $S_d$. In this paper we first give a decomposition $W_{(d-1,1)}(V,\K)=\bigoplus_{i=1}^{d-1}W_{(d-1,1)}^i(V,\K)$. We finally prove the vanishing of the hyperdeterminant of any $F\in(\bigoplus_{λ\ne(d),(d-1,1)})\oplus W_{(d-1,1)}^i(V,\K)$. This improves the result in [10] and [1], where the same result was proved without this new last summand. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06603 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the vanishing of the hyperdeterminant under certain symmetry conditions Arrondo, Enrique Tocino, Alicia Algebraic Geometry Given a vector space $V$ over a field $\K$ whose characteristic is coprime with $d!$, let us decompose the vector space of multilinear forms $V^*\otimes\overset{\text(d)}{\ldots}\otimes V^*=\bigoplus _λW_λ(X,\K)$ according to the different partitions $λ$ of $d$, i.e. the different representations of $S_d$. In this paper we first give a decomposition $W_{(d-1,1)}(V,\K)=\bigoplus_{i=1}^{d-1}W_{(d-1,1)}^i(V,\K)$. We finally prove the vanishing of the hyperdeterminant of any $F\in(\bigoplus_{λ\ne(d),(d-1,1)})\oplus W_{(d-1,1)}^i(V,\K)$. This improves the result in [10] and [1], where the same result was proved without this new last summand. |
| title | On the vanishing of the hyperdeterminant under certain symmetry conditions |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2407.06603 |