Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.20007 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912503212015616 |
|---|---|
| author | Gupta, Shubhanshi |
| author_facet | Gupta, Shubhanshi |
| contents | For a prime $p,$ let $\mathbb{F}_q$ be a finite extension of $\mathbb{F}_p.$ The restriction of an irreducible mod $p$ representation of $\text{GL}_2(\mathbb{F}_q)$ to its subgroup $\text{GL}_2(\mathbb{F}_p)$ can be seen as a tensor product of irreducible representations of $\text{GL}_2(\mathbb{F}_p).$ In this paper, we study the restriction of some of these representations of $\text{GL}_2(\mathbb{F}_q)$ to $\text{GL}_2(\mathbb{F}_p),$ for $q=p^2$ and $p^3$ using elementary tools and give explicit socle filtration when $q=4.$ We prove that when $q=p^2,$ a special class of representations of $\text{GL}_2(\mathbb{F}_q)$ are distinguished by suitable characters of $\text{GL}_2(\mathbb{F}_p).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20007 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the restriction of some irreducible mod $p$ representations Gupta, Shubhanshi Representation Theory For a prime $p,$ let $\mathbb{F}_q$ be a finite extension of $\mathbb{F}_p.$ The restriction of an irreducible mod $p$ representation of $\text{GL}_2(\mathbb{F}_q)$ to its subgroup $\text{GL}_2(\mathbb{F}_p)$ can be seen as a tensor product of irreducible representations of $\text{GL}_2(\mathbb{F}_p).$ In this paper, we study the restriction of some of these representations of $\text{GL}_2(\mathbb{F}_q)$ to $\text{GL}_2(\mathbb{F}_p),$ for $q=p^2$ and $p^3$ using elementary tools and give explicit socle filtration when $q=4.$ We prove that when $q=p^2,$ a special class of representations of $\text{GL}_2(\mathbb{F}_q)$ are distinguished by suitable characters of $\text{GL}_2(\mathbb{F}_p).$ |
| title | On the restriction of some irreducible mod $p$ representations |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2405.20007 |