Seminormal basis for the cyclotomic Hecke algebra of type $G(r,p,n)$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917806160740352 |
|---|---|
| author | Hu, Jun Wang, Shixuan |
| author_facet | Hu, Jun Wang, Shixuan |
| contents | The cyclotomic Hecke algebra $H_{r,p,n}$ of type $G(r,p,n)$ (where $r=pd$) can be realized as the $σ$-fixed point subalgebra of certain cyclotomic Hecke algebra $H_{r,n}$ of type $G(r,1,n)$ with some special cyclotomic parameters, where $σ$ is an automorphism of $H_{r,n}$ of order $p$. In this paper we prove a number of rational properties on the $γ$-coefficients arising in the construction of the seminormal basis for the semisimple Hecke algebra $H_{r,n}$. Using these properties, we construct a seminormal basis for the semisimple Hecke algebra $H_{r,p,n}$ in terms of the seminormal basis for the semisimple Hecke algebra $H_{r,n}$. The proof relies on some careful and subtle study on some rational and symmetric properties of some quotients and/or products of $γ$-coefficients of $H_{r,n}$. As applications, we obtain an explicit basis for the center $Z(H_{r,p,n})$ and an explicit basis for the $σ$-twisted $k$-center $Z(H_{r,n})^{(k)}$ of $H_{r,n}$ for each $k\in\mathbb{Z}/p\mathbb{Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13158 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Seminormal basis for the cyclotomic Hecke algebra of type $G(r,p,n)$ Hu, Jun Wang, Shixuan Representation Theory The cyclotomic Hecke algebra $H_{r,p,n}$ of type $G(r,p,n)$ (where $r=pd$) can be realized as the $σ$-fixed point subalgebra of certain cyclotomic Hecke algebra $H_{r,n}$ of type $G(r,1,n)$ with some special cyclotomic parameters, where $σ$ is an automorphism of $H_{r,n}$ of order $p$. In this paper we prove a number of rational properties on the $γ$-coefficients arising in the construction of the seminormal basis for the semisimple Hecke algebra $H_{r,n}$. Using these properties, we construct a seminormal basis for the semisimple Hecke algebra $H_{r,p,n}$ in terms of the seminormal basis for the semisimple Hecke algebra $H_{r,n}$. The proof relies on some careful and subtle study on some rational and symmetric properties of some quotients and/or products of $γ$-coefficients of $H_{r,n}$. As applications, we obtain an explicit basis for the center $Z(H_{r,p,n})$ and an explicit basis for the $σ$-twisted $k$-center $Z(H_{r,n})^{(k)}$ of $H_{r,n}$ for each $k\in\mathbb{Z}/p\mathbb{Z}$. |
| title | Seminormal basis for the cyclotomic Hecke algebra of type $G(r,p,n)$ |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2410.13158 |