Related papers: Divisibility of Character Values of Representation…

Let $q$ be a prime power, and $d$ a positive integer. We study the proportion of irreducible characters of $\mathrm{GL}(n,q)$ whose values evaluated on a fixed matrix $g$ are divisible by $d$. As $n$ approaches infinity, this proportion…

In [APS], the authors characterize the partitions of $n$ whose corresponding representations of $S_n$ have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups $W$. Namely, given a nontrivial…

A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some…

The paper relates character value of an irreducible representation of a compact connected Lie group at certain elements of finite order with the dimension of a representation on another group, up to some precise constants, which all have…

In this work we study representations of certain Coxeter groups to obtain some properties of the corresponding reflection groups.

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…

Let $G$ be a finite group and $d$ the degree of a complex irreducible character of $G$, then write $|G|=d(d+e)$ where $e$ is a nonnegative integer. We prove that $|G|\leq e^4-e^3$ whenever $e>1$. This bound is best possible and improves on…

Let $p$ be a prime. For $p=2$, the fields of values of the complex irreducible characters of finite groups whose degrees are not divisible by $p$ have been classified; for odd primes $p$, a conjectural classification has been proposed. In…

We give a new formula for the values of an irreducible character of the symmetric group S_n indexed by a partition of rectangular shape. Some observations and a conjecture are given concerning a generalization to arbitrary shapes.

In this paper we study (asymptotic) properties of the $*$-distribution of irreducible characters of finite quantum groups. We proceed in two steps, first examining the representation theory to determine irreducible representations and their…

A complex irreducible character of a finite group G is said to be p-constant, for some prime p dividing the order of G, if it takes constant value at the set of p-singular elements of G. In this paper we classify irreducible p-constant…

For any irreducible character $\chi$ of a finite group $G$, let $\theta(\chi)$ denote the proportion of elements $g\in G$ for which $\chi(g)$ is either zero or a root of unity. Then for any $L\in[1/2,1]$ and any $\epsilon>0$, there exists…

We investigate a beautiful conjecture of T. Wilde on character values and element orders of finite groups. We reduce it to a statement on nearly simple groups that can be checked ``prime by prime". For these groups, we show that a strong…

The character codegree of an irreducible character of a finite group $G$ is given by the index of its kernel in $G$ upon the character degree. We compute the codegrees of irreducible characters of VZ and Camina $p$-groups, and also obtain…

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…

Proving a conjecture of Miller, we show that as $n$ tends to infinity almost all entries in the character table of $S_n$ are divisible by any given prime power. This extends our earlier work which treated divisibility by primes.

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,…

Let \chi be an irreducible character of the finite group G. If g is an element of G and \chi(g) is not zero, then we conjecture that the order of g divides |G|/\chi(1). The conjecture is a generalization of the classical fact that…

Let N be a normal subgroup of a finite group G and consider the set cd(G|N) of degrees of irreducible characters of G whose kernels do not contain N. A number of theorems are proved relating the set cd(G|N) to the structure of N. For…

Let W be an infinite irreducible Coxeter group with (s_1, ..., s_n) the simple generators. We give a simple proof that the word s_1 s_2 ... s_n s_1 s_2 >... s_n ... s_1 s_2 ... s_n is reduced for any number of repetitions of s_1 s_2 >...…