Exponential matrices, $\mathbb{G}_a$-actions on projective spaces and modular representations of elementary abelian $p$-groups
Ryuji Tanimoto
TL;DR
The paper establishes explicit, compatible bijections linking exponential matrices over $k[T]$, $\,\mathbb{G}_a$-actions on $\mathbb{P}^{n-1}$, and modular representations of elementary abelian $p$-groups in positive characteristic. It builds a unified framework by (i) proving a chain of equivalences among six natural objects (exponential matrices, Hopf-algebra maps, algebraic-group maps to ${\rm GL}(n,k)$, ${\rm PGL}(n,k)$, ${\rm Aut}(\mathbb{P}^{n-1})$, and Ga-actions), and (ii) classifying exponential matrices via $p$-nilpotent, commuting matrices $N_i$ and their exponential data, culminating in a bijection between $\mathbb{E}_{\min}(n,k)$ and ${\rm Mat}(n,k[T])^E$. A key outcome is a description of $\mathbb{E}_{\ge0}(n,k)$ as $\mathbb{E}_{\min}(n,k)\times \mathbb{Z}_{\ge0}$, i.e., a trivial $\mathbb{Z}_{\ge0}$-fibration over the minimal data. Overall, the work provides a concrete, computational bridge between linear-algebraic data and geometric/group-action perspectives, enabling unified analysis across these areas in positive characteristic.
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 ${\rm Mat}(n, k[T])^E \times \mathbb{Z}_{\geq 0}$.