Let G0 be a connected unipotent group over a finite field Fq, and let G = G0 ⊗Fq Fq, equipped with the Frobenius endomorphism Frq : G −→ G. For every character sheaf M on G such that Frq M ∼= M , we prove that M comes from an irreducible perverse sheaf M0 on G0 such that M0 is pure of weight 0 (as an `-adic complex) and for each integer n ≥ 1 the “trace of Frobenius” function tM0⊗FqFqn on G0(Fq...

Let G be a finite group and H be a subgroup. We say that H is degree homogeneous if, for each χ ∈ Irr(G), all the irreducible constituents of the restriction χH have the same degrees. Subgroups which are either normal or abelian are obvious examples of degree homogeneous subgroups. Following a question of E.M. Zhmud’, we investigate general properties of such subgroups. It appears unlikely that...

The total character τ of a finite group G is defined as the sum of all the irreducible characters of G. K. W. Johnson asks when it is possible to express τ as a polynomial with integer coefficients in a single irreducible character. In this paper, we give a complete answer to Johnson’s question for all finite dihedral groups. In particular, we show that, when such a polynomial exists, it is uni...

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph ∆(G) is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ∈ cd(G). It is shown using the degree graphs of the finite simple groups that if G is a nonsolvable group, then the diameter of ∆(G) is at...

The present paper is a survey of a simple combinatorial model for the irreducible characters of complex semisimple Lie algebras, and, more generally, of complex symmetrizable Kac-Moody algebras. This model, which will be referred to as the alcove path model, can be viewed as a discrete counterpart to the Littelmann path model. It allows us to give character formulas and a LittlewoodRichardson r...

Let G be a finite group and let p prime number. We prove that if $$\chi \in {{{\text {Irr}}}}_{p'}(G)$$ $${\text {Ker}}\chi $$ does not have solvable normal p-complement then there exists $$\psi such (1)>\chi (1)$$ {Ker}}\psi <{\text . This is $$p'$$ -version of classical theorem Broline Garrison. As consequence, we obtain results on p-parts character codegrees.