Exponential matrices, ${\mathbb{G}}_a$-actions on projective spaces and modular representations of elementary abelian $p$-groups

Abstract

Let $k$ be an algebraically closed field of positive characteristic $p$ and let ${\mathbb{G}}_a$ denote the additive group of $k$. Let $n\ge 1$ and let $\mathrm{Mat}(n,k[T]{)}^E$ denote the set of all exponential matrices of $\mathrm{Mat}(n,k[T])$. Let ${\mathbb{E}}_{\ge 0}(n,k)$ denote the set of all group homomorphisms from $(\mathbb{Z}/p\mathbb{Z}{)}^r$ to $\mathrm{GL}(n,k)$, where $r$ ranges over all non-negative integers. In the first, we show that there exists a one-to-one correspondence between the set $\mathrm{Mat}(n,k[T]{)}^E$ and the set of all ${\mathbb{G}}_a$-actions on ${\mathbb{P}}^{n-1}$. In the second, we show that there exists a one-to-one correspondence between ${\mathbb{E}}_{\ge 0}(n,k)$ and the set $\mathrm{Mat}(n,k[T]{)}^E\times {\mathbb{Z}}_{\ge 0}$.