Related papers: Counting real characters after Gallagher

We are interested in determining the bound of the average of the degrees of the irreducible characters whose degrees are not divisible by some prime $p$ that guarantees a finite group $G$ of odd order is $p$-nilpotent. We find a bound that…

For any finite group $G$, Thompson proved that, for each $\chi\in {\rm Irr}(G)$, $\chi(g)$ is a root of unity or zero for more than a third of the elements $g\in G$, and Gallagher proved that, for each larger than average class $g^G$,…

This paper computes the irreducible characters of the alternating Hecke algebras, which are deformations of the group algebras of the alternating groups. More precisely, we compute the values of the irreducible characters of the semisimple…

Let $\Gamma$ be a finite simple graph. If for some integer $n\geqslant 4$, $\Gamma$ is a $K_n$-free graph whose complement has an odd cycle of length at least $2n-5$, then we say that $\Gamma$ is an $n$-exact graph. For a finite group $G$,…

When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the…

Let $G$ be a finite group and $V$ be a finite $G$--module. We present upper bounds for the cardinalities of certain subsets of $\Irr(GV)$, such as the set of those $\chi\in\Irr(GV)$ such that, for a fixed $v\in V$, the restriction of $\chi$…

The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra $\mathfrak{gl}(m|n)$ over $\C$ was solved a few years ago by V. Serganova. In this article, we present an entirely…

Let $G$ be a finite group and $\mathrm{Irr}(G)$ be the set of irreducible characters of $G$. The codegree of an irreducible character $\chi$ of the group $G$ is defined as $\mathrm{cod}(\chi)=|G:\mathrm{ker}(\chi)|/\chi(1)$. In this paper,…

We determine the finite groups whose real irreducible representations have different degrees.

Let $G$ be a real reductive group in Harish-Chandra's class. We derive some consequences of theory of coherent continuation representations to the counting of irreducible representations of $G$ with a given infinitesimal character and a…

Using the construction by Bencs and T\'{o}th of invariant random subgroups on weakly branch groups acting on regular rooted trees we produce uncountably many indecomposable characters on these groups. In fact, we study three types of…

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.…

For an irreducible character $\chi$ of a finite group $G$, the codegree of $\chi$ is defined as $|G:\ker(\chi)|/\chi(1)$. In this paper, we determine finite nonsolvable groups with exactly three nonlinear irreducible character codegrees,…

Let G be a finite solvable group and $\chi\in \Irr(G)$ be a faithful character. We show that the derived length of G is bounded by a linear function of the number of distinct irreducible constituents of $\chi\bar{\chi}$. We also discuss…

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.

For every countable group $G$, there are $2^{\omega}$ distinct classes of coarsely equivalent subsets of $G$.

A finite group $G$ is $normally ~monomial$ if all its irreducible characters are induced from linear characters of normal subgroups of $G$. In this paper, we determine all possible irreducible character degree sets of normally monomial…

We give an algorithm for computing the irreducible admissible representations of a real reductive group with regular integral infinitesimal character. This algorithm has been implemented on a computer, as part of the Atlas of Lie Groups and…

We define the position of an irreducible complex character of a finite group as an alternative to the degree. We then use this to define three classes of groups: PR-groups, IPR-groups and weak IPR-groups. We show that IPR-groups and weak…

If a group $G$ is $\pi$-separable, where $\pi$ is a set of primes, the set of irreducible characters $\operatorname{B}_{\pi}(G) \cup \operatorname{B}_{\pi'}(G)$ can be defined. In this paper, we prove that there are variants of some…