Real characters in blocks

TL;DR

This paper investigates real versions of key conjectures in block theory, proving specific cases for 2-blocks and characterizing classes related to defect groups.

Contribution

It introduces the real versions of Brauer's, Olsson's, and Eaton's conjectures and proves Eaton's conjecture for certain 2-blocks, also characterizing G-classes and related structures.

Findings

Proved the real Eaton's conjecture for 2-blocks with cyclic defect groups.

Characterized G-classes, real, and rational G-classes of defect groups.

Established conditions for principal 2-blocks with trivial real core.

Abstract

We consider real versions of Brauer's k(B) conjecture, Olsson's conjecture and Eaton's conjecture. We prove the real version of Eaton's conjecture for 2-blocks of groups with cyclic defect group and for the principal 2-blocks of groups with trivial real core. We also characterize G-classes, real and rational G-classes of the defect group of a block B.