Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.07939 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908584319647744 |
|---|---|
| author | Elmer, Jonathan Kadr, Kazal |
| author_facet | Elmer, Jonathan Kadr, Kazal |
| contents | Let $p>0$ be a prime, $k$ a field of characteristic $p$ and $G$ and elementary abelian $p$-group of order $q = p^n$. Let $W$ be an indecomposable $kG$-module of dimension 2 and define $V_i=S^{i-1}(W^*)$ for each $i=1 \ldots q$. We show that $V_2 \otimes V_i \cong V_{i+1} \oplus V_{i-1}$ provided $i$ is not divisible by $p$, and that $V_2 \otimes V_p$ is indecomposable if $n>1$. Our results generalise results of Almkvist and Fossum for representations of cyclic groups of order $p$. We show how our results give formulae for the direct sum decomposition of $V_i \otimes V_j$ for $i<p$ and $j<j$ modulo summands projective to $\bigoplus_{r=0}^{p-1}V_{rp}$ and conjecture that these formulae extend to the case $i<q$ and $j<q$. We provide some evidence for our conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07939 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some formulae relating modular representations of elementary abelian $p$-groups Elmer, Jonathan Kadr, Kazal Representation Theory 20C20 Let $p>0$ be a prime, $k$ a field of characteristic $p$ and $G$ and elementary abelian $p$-group of order $q = p^n$. Let $W$ be an indecomposable $kG$-module of dimension 2 and define $V_i=S^{i-1}(W^*)$ for each $i=1 \ldots q$. We show that $V_2 \otimes V_i \cong V_{i+1} \oplus V_{i-1}$ provided $i$ is not divisible by $p$, and that $V_2 \otimes V_p$ is indecomposable if $n>1$. Our results generalise results of Almkvist and Fossum for representations of cyclic groups of order $p$. We show how our results give formulae for the direct sum decomposition of $V_i \otimes V_j$ for $i<p$ and $j<j$ modulo summands projective to $\bigoplus_{r=0}^{p-1}V_{rp}$ and conjecture that these formulae extend to the case $i<q$ and $j<q$. We provide some evidence for our conjecture. |
| title | Some formulae relating modular representations of elementary abelian $p$-groups |
| topic | Representation Theory 20C20 |
| url | https://arxiv.org/abs/2510.07939 |