Kronecker coefficients and Harrison centres of the representation ring of the symmetric group
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913883896152064 |
|---|---|
| author | Sun, Jia-Cheng Zhang, Chi Zhu, Haoran |
| author_facet | Sun, Jia-Cheng Zhang, Chi Zhu, Haoran |
| contents | We present a computational approach to studying the structure of the representation ring of the symmetric group in dimension six. The Kronecker coefficients and all power formulae of irreducible representations of $S_6$ are computed using the character theory of finite groups. In addition, considering direct sum decomposition of tensor products of different irreducible representations of $S_6$, we characterise generators of the representation ring $\mathcal{R}(S_6)$, show that its unit group $U(\mathcal{R}(S_6))$ is a Klein four-group, and related results on the structure of primitive idempotents. Furthermore, we introduce the Harrison centre theory to study the representation ring and show that the Harrison centre of the cubic form induced by the generating relations of $\mathcal{R}(S_6)$ is isomorphic to itself. Finally, we conclude with some open problems for future consideration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18152 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kronecker coefficients and Harrison centres of the representation ring of the symmetric group Sun, Jia-Cheng Zhang, Chi Zhu, Haoran Representation Theory Rings and Algebras 20C30, 20C15, 11E76, 05E10 We present a computational approach to studying the structure of the representation ring of the symmetric group in dimension six. The Kronecker coefficients and all power formulae of irreducible representations of $S_6$ are computed using the character theory of finite groups. In addition, considering direct sum decomposition of tensor products of different irreducible representations of $S_6$, we characterise generators of the representation ring $\mathcal{R}(S_6)$, show that its unit group $U(\mathcal{R}(S_6))$ is a Klein four-group, and related results on the structure of primitive idempotents. Furthermore, we introduce the Harrison centre theory to study the representation ring and show that the Harrison centre of the cubic form induced by the generating relations of $\mathcal{R}(S_6)$ is isomorphic to itself. Finally, we conclude with some open problems for future consideration. |
| title | Kronecker coefficients and Harrison centres of the representation ring of the symmetric group |
| topic | Representation Theory Rings and Algebras 20C30, 20C15, 11E76, 05E10 |
| url | https://arxiv.org/abs/2407.18152 |