Dear Vince, dear Forum,
On Fri, Jun 28, 2019 at 03:11:49PM -1000, Vince Giambalvo wrote:
> Does anyone have thoughts on how to construct real character tables for small
> finite groups. It seems as if this is
> something that someone has already done a long time ago. I suppose it is
> possible to construct the “real” conjugacy
> classes (those containing both an element and its inverse), but if someone
> has an easier suggestion or has already done it,
> I would appreciate the help.
All the information you need is already in the ordinary character
tables. You merely need to know which real-valued irreducible characters \chi
are
quaternionic, by computing Schur index (which should be 2 in the
quaternionic case),
or equivalently, the sign of the sum \sum_{g in G} \chi(g^2) (which
should be negative in the quaternionic case, cf. Prop 39 in Sect. 13.2 of [1]).
The latter is an easy computation in GAP, using the power maps and the
values of \chi.
[1] J.-P. Serre, "Linear representations of finite groups", GTM vol. 42,
Springer, 1977.
Hope this helps,
Dima
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