Representations and characters of finite groups

This article examines lower bounds for the representation growth of finitely generated (particularly profinite and pro-p) groups. It also considers the related question of understanding the maximal multiplicities of character degrees in finite groups, and in particular simple groups.

These are partial lecture notes from the fifteen Ess\'en Lectures for graduate students at Uppsala University given (in four days!) in June 2013.

For a finite group $G$, the representation dimension is the smallest integer realizable as the degree of a complex faithful representation of $G$. In this article, we compute representation dimension for some $p$-groups, their direct products, and groups with certain conditions on nonlinear irreducible characters. We also make similar computations for the smallest integer realizable as the degree of an irreducible complex faithful representation of $G$, if one exists. In the ...

This is a survey article on the theory of finite complex reflection groups. No proofs are given but numerous references are included.

In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.

These informal notes concern some basic themes of harmonic analysis related to representations of groups.

We give parameterizations of the irreducible representations of finite groups of Lie type in their defining characteristic.

This is an expanded version of my Shaw Prize Lecture delivered at the Chinese University of Hong Kong.

By generalizing Frobenius' polynomial method to good partition algebra, we will develop new character theories for a finite group $G$. A uniform defining equations are derived for these kinds of character theories. The new character theories leads to various factorizations of the group determinant. We will show that these new character theories are equivalent to the Frobenius polynomials of the correspondent good partition algebras. In particular, the character table of a fin...

In this work, we classify all finite groups such that for every field extension F of \mathbb{Q}, F is the field of values of at most 3 irreducible characters.