In this paper we prove some mainly asymptotic results concerning the irreducible character degrees of .nite groups of Lie type. Applications are given to the study of the mixing time of random walks on these groups, with certain conjugacy classes as generating sets. In various situations we show that the mixing time is 2; this seems to be the .rst determination of an exact bounded mixing time f...

For any partition, the corresponding Specht module is the sub module of permutation module spanned by the poly-tabloids. The Specht modules for a partition of a positive integer n form a complete list of irreducible representations of permutation group of degree n. The Semi-Standard Young Tableau (SSYT), on n -symbols are one type of combinatorial objects occur naturally in many computational p...

Farey sequences of irreducible fractions between 0 and 1 can be related to graph constructions known as Farey graphs. These graphs were first introduced by Matula and Kornerup in 1979 and further studied by Colbourn in 1982 and they have many interesting properties: they are minimally 3-colorable, uniquely Hamiltonian, maximally outerplanar and perfect. In this paper we introduce a simple gener...

We obtain estimates on the character of the cohomology of an S-equivariant holomorphic vector bundle over a Kähler manifold M in terms of the cohomology of the Lerman symplectic cuts and the symplectic reduction of M . In particular, we prove and extend inequalities conjectured by Wu and Zhang [12]. The proof is based on constructing a flat family of complex spaces Mt (t ∈ C) such that Mt is is...

The Lie algebra of a compact semisimple Lie group G is determined by the degrees of the irreducible representations of G. However, two different groups can have the same representation degrees.

The minimal faithful degree of a finite group G, denoted by μ(G), is the least non-negative integer n such that G embeds inside Sym(n). In this article we calculate the minimal faithful permutation degree for all of the irreducible Coxeter groups.

LetG be a permutation group of degree n. The connection between themultiple transitivity of G and the irreducibility of certain irreducible characters of of the symmetric group Sn when restricted to G goes back to Frobenius, and a proof of his classical result can be found in Tsuzuku 1 . In 2 , Saxl strengthens this result, limiting the irreducibility hypothesis to the characters χ n−k,k G , a ...

The natural character π of a finite transitive permutation group G has the form 1G + θ where θ is a character which affords a rational representation of G. We call G a QI-group if this representation is irreducible over Q. Every 2-transitive group is a QI-group, but the latter class of groups is larger. It is shown that every QI-group is 3/2transitive and primitive, and that it is either almost...

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 questions.