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...

where χ(1) denotes the dimension of the character χ and (n)k = n(n − 1) · · · (n − k + 1). Thus [8, (7.6)(ii)][12, p. 349] χ(1) is the number f of standard Young tableaux of shape λ. Identify λ with its diagram {(i, j) : 1 ≤ j ≤ λi}, and regard the points (i, j) ∈ λ as squares (forming the Young diagram of λ). We write diagrams in “English notation,” with the first coordinate increasing from to...

We study Shintani lifting of real-valued irreducible characters of finite reductive groups. In particular, if G is a connected reductive group defined over Fq, and ψ is an irreducible character of G(Fqm) which is the lift of an irreducible character χ of G(Fq), we prove ψ is real-valued if and only if χ is real-valued. In the case m = 2, we show that if χ is invariant under the twisting operato...

INTRODUCTION AND NOTATION: As usual, for G a finite group, we denote the set of non-identity elements of G by G, the number of conjugacy classes of G by k(G) and the set of complex irreducible characters ofG by Irr(G). When H is a non-trivial subgroup of G, we let k+(G,H) denote the number of irreducible characters of G which do not vanish identically on H (and we remark that k+(G,H) = k(G) whe...