A Local Conjecture on Brauer Character Degrees of Finite Groups

Recently, a new conjecture on the degrees of the irreducible Brauer characters of a finite group was presented in [16]. In this paper we propose a ’local’ version of this conjecture for blocks B of finite groups, giving a lower bound for the maximal degree of an irreducible Brauer character belonging to B in terms of the dimension of B and well-known invariants like the defect and the number of irreducible Brauer characters. We also propose a weaker version of this conjecture for which a slight reformulation leads to interesting open questions about traces of Cartan matrices of blocks. We then show that the strong conjecture is true for blocks with one simple module, blocks of p-solvable groups and blocks with cyclic defect groups. It also holds for many further examples of blocks of sporadic groups, symmetric groups or groups of Lie type. We also show that the weak conjecture is true for blocks of tame representation type. Introduction Representation theory of finite groups is an area which is characterized by an enormous number of important long-standing open problems. Many of them deal with ordinary or modular characters, so character theory still is one of the most important and fundamental parts of representation theory. The key aim in this area is to find intimate relations between the numerical invariants given by characters and the structure of the group. Actually, R. Brauer’s main motivation for introducing modular characters was to study the structure of finite simple groups. In this paper we are going to study some of the most important numerical invariants in modular representation theory, namely degrees of irreducible Brauer characters. The main aim is to propose a conjecture relating the degrees of the irreducible characters belonging to a block to some well-known invariants of the block. This can be considered as a block version of a conjecture recently presented in [16] for finite groups. In order to state the conjecture we need to introduce some notation. Let G be a finite group and let p be a prime dividing the order of G. Moreover, let B be a p-block of G with defect group D where the underlying field of characteristic p is assumed to be algebraically closed. Furthermore let Irrp(B) = {φ1, . . . , φl(B)} denote the set of irreducible Brauer characters belonging to B and Irrp(G) the set of all irreducible Brauer characters of G. In [16] the second author proposed the following conjecture relating the degrees of the irreducible Brauer characters of a finite group G to the p′-part of the order of G (i.e. the largest divisor of the order of G not divisible by p). Global Conjecture [16]: For any finite group G we have |G|p′ ≤ ∑