Jordan blocks of unipotent elements in some irreducible representations of classical groups in good characteristic

TL;DR

This paper characterizes the Jordan block sizes of unipotent elements acting on certain irreducible modules of classical groups over algebraically closed fields of good characteristic, using recursive formulas for related tensor constructions.

Contribution

It provides explicit descriptions of Jordan block sizes for unipotent elements on key modules, extending understanding of their structure in classical groups.

Findings

Jordan block sizes are described in terms of tensor, exterior, and symmetric squares.

Recursive formulas for these block sizes are utilized.

Results apply to modules arising as composition factors of natural tensor constructions.

Abstract

Let $G$ be a classical group with natural module $V$ over an algebraically closed field of good characteristic. For every unipotent element $u$ of $G$, we describe the Jordan block sizes of $u$ on the irreducible $G$-modules which occur as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$. Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of $u$, for which recursive formulae are known.