Decomposition of exterior and symmetric squares in characteristic two

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∈sl(V), we describe the Jordan normal form of e on the sl(V)-modules ∧2(V) and S2(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∈SL(V), the Jordan normal form of u on ∧2(V) and S2(V). A recursive formula for the Jordan block sizes of u on ∧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 S2(V).