Is there some easy way to identify from the character table record
which columns are inverses of which other columns? I'd like to be able
to do this to, say, compute class multiplication constants
automatically (by a pre-defined function) from the table. Everything
else needed seems to be readily available in the record.
One could figure this out from the power maps in the record, but that
might be messy to do in a function.
Alternatively, one could just use the complex conjugate of a column,
but I don't even see how to do that in a function. Presumably this is
somewhere in the automorphisms of the table, but how would I tell which
automorphism is complex conjugation? Or identify that the table is
real, so that complex conjugation does nothing, and is therefore not
among the listed automorphisms.
Is it efficient just to use column orthogonality? The one column not
orthogonal to a given one is its inverse.
Probably I'm just missing something obvious here.