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 by W. Willems. In this paper we propose a `local' version of this conjecture for blocks $B$ of finite groups, giving a lower bound for $\sum \varphi(1)^2$ where the sum runs through the set of irreducible Brauer characters of $B$ in terms of invariants of $B$. A slight reformulation leads to interesting open questions about traces of Cartan matrices of blocks.

We show that the local 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. Finally we prove that the conjecture is true for blocks of tame representation type.