On Sat, Jun 29, 2019 at 08:41:52AM +0100, dmitrii.pasech...@cs.ox.ac.uk wrote:
> 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.
At this point you merely need to take the real characters in the character
table, add to them
\chi+\overline{\chi} for every complex character, and replace each
quaternionic character \chi with 2\chi.
After this procedure, some columns (i.e. conjugacy classes) of the table
might become identical, so you'd merge them.
>
> Hope this helps,
> Dima
>
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