"How to count the irreducible characters of a finite group that have a given
Frobenius-Schur indicator"

Dr John Murray

Abstract:
It is well known that for a finite group, the number of irreducible complex
characters is the same as the number of conjugacy classes. Analogously, the
number of real irreducible characters coincides with the number of real
conjugacy classes. Now each real irreducible character has Frobenius-Schur
indicator \pm 1. Richard Brauer posed the problem of determining the number
with indicator -1, using group theoretic information. This remains an open
problem.

In this talk I outline some recent progress on the problem that relies on
Brauer's theory of blocks of characters. Specifically, we answer the problem
for 2-blocks with finite or domestic representation type, and indicate our
ongoing efforts to complete the solution for blocks with dihedral defect
group.