Decomposition of exterior and symmetric squares in characteristic two

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 $e \in \mathfrak{sl}(V)$,wedescribetheJordannormalformof$e$onthe$\mathfrak{sl}(V)$-modules$\wedge^2(V)$and$S^2(V)$\mathrm{.}Inthecasewhere$e$ 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 $u\in \mathrm{SL}(V)$, 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 $S^2(V)$.