Totally orthogonal finite simple groups

Abstract

We prove that if G is a finite simple group, then all irreducible complex representations of G may be realized over the real numbers if and only if every element of G may be written as a product of two involutions in G. This follows from our result that if q is a power of 2, then all irreducible complex representations of the orthogonal groups O ± (2n, Fq) may be realized over the real numbers. We also obtain generating functions for the sums of degrees of several sets of unipotent characters of finite orthogonal groups, and we obtain a twisted version of our main result for a broad family of finite classical groups. 2010

Some Reality Properties of Finite Simple Orthogonal Groups

“…Proof. If q is a power of 2, then this statement is proven in [20,Theorems 8.3 and 8.6] and in [19,Theorem 4.2] (noting that SO(2m+1, q) ∼ = Sp(2m, q) when q is a power of 2). Thus we now assume that q is the power of an odd prime.…”

“…In particular, in some classes of finite groups there is a connection between strongly real classes and the irreducible complex characters which are afforded by a real representation. For example, it is proven in [20] that for a finite simple group, all conjugacy classes are strongly real if and only if all irreducible complex characters can be afforded by a real representation. In this paper we gather evidence that for finite simple orthogonal groups, all real classes are strongly real if and only if all real-valued irreducible complex characters can be afforded by real representations.…”