Some formulae relating modular representations of elementary abelian $p$-groups
Jonathan Elmer, Kazal Kadr
TL;DR
The paper studies modular representations of elementary abelian $p$-groups via symmetric powers and tensor products. It constructs a family of indecomposable modules $V_i=S^{i-1}(W)^*$ from a faithful 2-dimensional $kG$-module $W$ and proves a sharp two-term tensor product formula $V_2\otimes V_i\cong V_{i+1}\oplus V_{i-1}$ when $p$ does not divide $i$, with $V_2\otimes V_p$ indecomposable when $n>1$. Building on Almkvist–Fossum, the authors describe decompositions of $V_i\otimes V_j$ modulo summands projective to $M=\bigoplus_{r=1}^{p^{n-1}} V_{rp}$ and present stable analogues and conjectures extending these formulas to the full range $i<q$, $j<q$ with supporting evidence. The work develops the Heller shift for these modules and situates the results within the stable module category, connecting to known results for cyclic groups and $\mathrm{SL}_2(\Bbbk)$ representations to provide a framework for future extensions.
Abstract
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.