Almost cyclic elements in Weil representations of finite classical groups

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 (that is, a square matrix M of size n over a field F with the property that there exists α∈F such that M is similar to $diag(\alpha \cdot {Id}_k,M_1)$, where M₁ is cyclic and 0≤k≤n). 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.

KEYWORDS:

• Classical groups
• eigenvalue multiplicities
• Weil representations

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20C15
• 20C20
• 20C33

Acknowledgments

We wish to thank Marco Antonio Pellegrini for having helped us with his skills in the use of the MAGMA and GAP packages for dealing with character tables and Brauer character tables, and for having provided efficient ad hoc routines for testing cyclicity and almost cyclicity of matrices.