Lower Bounds for Degrees of Irreducible Brauer Characters of Finite General Linear Groups

On Degrees of Irreducible Brauer Characters

Based on a large amount of examples, which we have checked so far, we conjecture that |G|p′ ≤ ∑ φ φ(1) 2 where p is a prime and the sum runs through the set of irreducible Brauer characters in characteristic p of the finite group G. We prove the conjecture simultaneously for p-solvable groups and groups of Lie type in the defining characteristic. In non-defining characteristics we give asymptot...

Bounds for Degrees and Sums of Degrees of Irreducible Characters of Some Classical Groups over Finite Fields

The goal of this note (which is incorporated and expanded in Chapter 5 of the author’s book “The large sieve and its applications” ) is to bound from above in a suitable manner the degree of irreducible representations, and the sum of the degrees of irreducible representations, of a group G`, which in applications is either between SL(r,F`) and GL(r,F`), or between Sp(2g,F`) and CSp(2g,F`). Mor...

On Characters of Irreducible Unitary Representations of General Linear Groups

Founding harmonic analysis on classical simple complex groups, I.M. Gelfand and M.A. Naimark in their classical book [GN] posed three basic questions: unitary duals, characters of irreducible unitary representations and Plancherel measures. In the case of reductive p-adic groups, the only series of reductive groups where unitary duals are known are general linear groups. In this paper we reduce...

Some connections between powers of conjugacy classes and degrees of irreducible characters in solvable groups

‎Let $G$ be a finite group‎. ‎We say that the derived covering number of $G$ is finite if and only if there exists a positive integer $n$ such that $C^n=G'$ for all non-central conjugacy classes $C$ of $G$‎. ‎In this paper we characterize solvable groups $G$ in which the derived covering number is finite‎.‎ 

On Finite Groups With Two Irreducible Character Degrees