Decomposition of exterior and symmetric squares in characteristic two

TL;DR

This paper characterizes the Jordan normal form of nilpotent and unipotent elements acting on the exterior and symmetric squares of a vector space over a field of characteristic two, providing explicit formulas and descriptions.

Contribution

It provides a detailed description of the Jordan normal form for these actions, including a closed formula for regular nilpotent elements and an adaptation of existing recursive formulas.

Findings

Explicit Jordan normal forms for nilpotent elements on exterior and symmetric squares.

Closed formula for regular nilpotent elements in characteristic two.

Adaptation of recursive formulas for unipotent elements' actions.

Abstract

Let $V$ be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element $e \in \mathfrak{sl}(V)$, we describe the Jordan normal form of $e$ on the $\mathfrak{sl}(V)$-modules $\wedge^2(V)$ and $S^2(V)$. In the case where $e$ is a regular nilpotent element, we are able to give a closed formula. We also consider the closely related problem of describing, for every unipotent element $u \in \operatorname{SL}(V)$, the Jordan normal form of $u$ on $\wedge^2(V)$ and $S^2(V)$. A recursive formula for the Jordan block sizes of $u$ on $\wedge^2(V)$ was given by Gow and Laffey (J. Group Theory 9 (2006), 659-672). We show that their proof can be adapted to give a similar formula for the Jordan block sizes of $u$ on $S^2(V)$.