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.

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

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

In the previous paper, we proposed a practical method of constructing explicitly representation groups $R(G)$ for finite groups $G$, and apply it to certain typical finite groups $G$ with Schur multiplier $M(G)$ containing prime number 3. In this paper, we construct a complete list of irreducible projective (or spin) representations of $G$ and compute their characters (called spin characters). It is a continuation of our study of spin representations in the cases where $M(G)$...

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.

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.