Two classes of minimal generic fundamental invariants for tensors
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2021
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910931792953344 |
|---|---|
| author | Li, Xin Zhang, Liping Xia, Hanchen |
| author_facet | Li, Xin Zhang, Liping Xia, Hanchen |
| contents | Motivated by the problems raised by Bürgisser and Ikenmeyer, we discuss two classes of minimal generic fundamental invariants for tensors of order 3. The first one is defined on $\otimes^3 \mathbb{C}^m$, where $m=n^2-1$. We study its construction by obstruction design introduced by Bürgisser and Ikenmeyer, which partially answers one problem raised by them. The second one is defined on $\mathbb{C}^{\ell m}\otimes \mathbb{C}^{mn}\otimes \mathbb{C}^{n\ell}$. We study its evaluation on the matrix multiplication tensor $\langle\ell,m,n\rangle$ and unit tensor $\langle n^2 \rangle$ when $\ell=m=n$. The evaluation on the unit tensor leads to the definition of Latin cube and 3-dimensional Alon-Tarsi problem. We generalize some results on Latin square to Latin cube, which enrich the understanding of 3-dimensional Alon-Tarsi problem. It is also natural to generalize the constructions to tensors of other orders. We illustrate the distinction between even and odd dimensional generalizations by concrete examples. Finally, some open problems in related fields are raised. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_07343 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Two classes of minimal generic fundamental invariants for tensors Li, Xin Zhang, Liping Xia, Hanchen Representation Theory Motivated by the problems raised by Bürgisser and Ikenmeyer, we discuss two classes of minimal generic fundamental invariants for tensors of order 3. The first one is defined on $\otimes^3 \mathbb{C}^m$, where $m=n^2-1$. We study its construction by obstruction design introduced by Bürgisser and Ikenmeyer, which partially answers one problem raised by them. The second one is defined on $\mathbb{C}^{\ell m}\otimes \mathbb{C}^{mn}\otimes \mathbb{C}^{n\ell}$. We study its evaluation on the matrix multiplication tensor $\langle\ell,m,n\rangle$ and unit tensor $\langle n^2 \rangle$ when $\ell=m=n$. The evaluation on the unit tensor leads to the definition of Latin cube and 3-dimensional Alon-Tarsi problem. We generalize some results on Latin square to Latin cube, which enrich the understanding of 3-dimensional Alon-Tarsi problem. It is also natural to generalize the constructions to tensors of other orders. We illustrate the distinction between even and odd dimensional generalizations by concrete examples. Finally, some open problems in related fields are raised. |
| title | Two classes of minimal generic fundamental invariants for tensors |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2111.07343 |