Related papers: Primitive Characters and Permutation Characters of…

Let G be a finite group of order n and V an irreducible representation over the complex numbers of dimension d. For some nonnegative number e, we have n=d(d+e). If e is small, then the character of V has unusually large degree. We fix e and…

In representation theory of finite groups an important role is played by irreducible characters of p-defect 0, for a prime p dividing the group order. These are exactly those vanishing at the p-singular elements. In this paper we generalize…

Let $G$ be a finite group. Denote by $\textrm{Irr}(G)$ the set of all irreducible complex characters of $G.$ Let $\textrm{cd}(G)=\{\chi(1)\;|\;\chi\in \textrm{Irr}(G)\}$ be the set of all irreducible complex character degrees of $G$…

Let us consider the group $G = < x,y \mid x^m = y^n>$ with $m$ and $n$ nonzero integers. In this paper, we study the variety of epresentations $R(G)$ and the character variety $X(G)$ in $SL(2,\C)$ of the group $G$,obtaining by elementary…

Let $G$ be a finite symmetric, general linear, or general unitary group defined over a field of characteristic coprime to $3$. We construct a canonical correspondence between irreducible characters of degree coprime to $3$ of $G$ and those…

We study definably primitive pseudo-finite permutation groups of finite $SU$-rank. We show that if $(G,X)$ is such a permutation group, then the rank of $G$ can be bounded in terms of the rank of $X$, providing an analogue of a theorem of…

If chi is an irreducible character of a finite group G then the support of chi is the subset of G on which chi does not vanish. In this note, we study the supports of characters of certain classes of p-groups (a p-group is a finite group of…

In this article we consider the connected component of the identity of $G$-character varieties of compact Riemann surfaces of genus $g > 0$, for connected complex reductive groups $G$ of type $A$ (e.g., $SL_n$ and $GL_n$). We show that…

Let $H$ be an extension of a finite group $Q$ by a finite group $G$. Inspired by the results of duality theorems for \'etale gerbes on orbifolds, we describe the number of conjugacy classes of $H$ that maps to the same conjugacy class of…

Let $p$ be a prime and let $\mathbb{C}$ be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of $\mathrm{GL}(p,\mathbb{C})$ up to conjugacy. That is, we give a complete and irredundant list of…

Let $\Gamma$ be a finitely generated group and $G$ a real form of $\mathrm{SL}_n(\mathbb{C})$. We propose a definition for the $G$-character variety of $\Gamma$ as a subset of the $\mathrm{SL}_n(\mathbb{C})$-character variety of $\Gamma$.…

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…

We classify all finite groups $G$ which possesses an element $x\in G$ such that every irreducible character of $G$ takes a root of unity value at $x$.

Let $G$ be a finite group, and let $\text{Irr}(G)$ denote the set of the irreducible complex characters of $G$. An element $g\in G$ is called a vanishing element of $G$ if there exists $\chi\in\text{Irr}(G)$ such that $\chi(g)=0$ (i.e., $g$…

Let $q$ be a prime power and $U$ the group of lower unitriangular matrices of order $n$ for some natural number $n$. We give a lower bound for the degrees of irreducible constituents of Andr\'{e}-Yan supercharacters and classify the…

Let $N$ be a normal subgroup of a finite group $G$. From a result due to Brauer, it can be derived that the character table of $G$ contains square submatrices which are induced by the $G$-conjugacy classes of elements in $N$ and the…

Together with David Schlang we computed the discriminants of the invariant Hermitian forms for all indicator $o$ even degree absolutely irreducible characters of the ATLAS groups supplementing the tables of orthogonal determinants computed…

We prove that if the average of the degrees of the irreducible characters of a finite group $G$ is less than 16/5, then $G$ is solvable. This solves a conjecture of I.M. Isaacs, M. Loukaki, and the first author. We discuss related…

A classical theorem on character degrees states that if a finite group has fewer than four character degrees, then the group is solvable. We prove a corresponding result on character values by showing that if a finite group has fewer than…

If $G$ is a finite classical group, linear or unitary in any characteristic, and orthogonal in odd characteristic, we give an approximate formula for $\chi(g)$ in which the error term is much smaller than the estimate, when $g\in G$ is an…