Almost cyclic elements in Weil representations of finite classical groups

TL;DR

This paper investigates the properties of almost cyclic matrices within Weil representations of finite classical groups, contributing to the broader goal of classifying irreducible representations of finite quasi-simple groups over algebraically closed fields.

Contribution

It specifically analyzes the occurrence of almost cyclic matrices in Weil representations of finite classical groups, advancing understanding of their structure and role in representation theory.

Findings

Identification of conditions under which Weil representations contain almost cyclic matrices.

Clarification of the role of almost cyclic matrices in the classification of irreducible representations.

Insights into the structure of Weil representations in relation to almost cyclic elements.

Abstract

This paper is a significant part of a general project aimed to classify all irreducible representations of finite quasi-simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix. (A square matrix $M$ is called almost cyclic if it is similar to a block-diagonal matrix with two blocks, such that one block is scalar and another block is a matrix whose minimum and characteristic polynomials coincide. Reflections and transvections are examples of almost cyclic matrices. The paper focuses on the Weil representations of finite classical groups, as there is strong evidence that these representations play a key role in the general picture.