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 $n \geq 1$ and let ${\rm Mat}(n, k[T])^E$ denote the set of all exponential matrices of ${\rm Mat}(n, k[T])$. Let $\mathbb{E}_{\geq 0}(n, k)$ denote the set of all group homomorphisms from $(\mathbb{Z}/p\mathbb{Z})^r$ to ${\rm GL}(n, k)$, where $r$ ranges over all non-negative integers. In the first, we show that there exists a one-to-one correspondence between the set ${\rm Mat}(n, k[T])^E$ 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 $\mathbb{E}_{\geq 0}(n, k)$ and the set ${\rm Mat}(n, k[T])^E \times \mathbb{Z}_{\geq 0}$.