Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with single non-trivial Jordan block

TL;DR

This paper characterizes when a non-trivial irreducible representation of a simple algebraic group of exceptional Lie type maps a unipotent element to a matrix with only one non-trivial Jordan block, focusing on the group G2.

Contribution

It provides a complete classification for exceptional groups, showing that only G2 with regular unipotent elements and small dimension representations have this property.

Findings

Only G2 with regular unipotent elements and dim ≤ 7 have single non-trivial Jordan block representations.

Classical groups' cases were previously classified by Suprunenko.

The result characterizes the structure of representations with minimal Jordan block complexity.

Abstract

In this paper we prove the following result. Let $G$ be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field $F$ of characteristic $p\geq 0$, and let $u\in G$ be a nonidentity unipotent element. Let $\phi$ be a non-trivial irreducible representation of $G$. Then the Jordan normal form of $\phi(u)$ contains at most one non-trivial block if and only if $G$ is of type $G_2$, $u$ is a regular unipotent element and $\dim \phi\leq 7$. Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a single non-trivial block were determined by I.D. Suprunenko (Unipotent elements of non-prime order in representations of the classical algebraic groups: two big Jordan blocks, J. Math. Sci. 199(2014), 350 -- 374.