Dimension statistics of representations of finite groups

Abstract

The paper studies the distribution of dimensions of irreducible representations of a finite group and compares them with the sizes of its conjugacy classes. We study these questions for a fixed reductive group over growing finite fields, for ${\mathrm{GL}}_n({\mathbb{F}}_q)$ where $q$ is fixed but $n$ varies, and for the symmetric group $S_n$ as $n$ varies. We also discuss finite nilpotent groups arising from Lie algebras over finite fields. As a consequence of Kirillov theory, we deduce that for ``selfdual'' nilpotent groups whose nilpotency class is less than the characteristic of the field, the squares of the dimensions of irreducible representations match the sizes of conjugacy classes, with multiplicity. These groups thus provide a large class of examples with this property.

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