How do I work with the character tables of Weyl groups in Sage to compute restrictions to parabolic subgroups?
The question is essentially what is in the title. To be more concrete, let's start with the Weyl group of type $E_6$. This contains a parabolic subgroup of type $D_5$. I know how to look at the character tables of these groups using sage: for instance
W=WeylGroup(["E",6]); ctE6=W.character_table(); ctE6
and I can do the same thing with $D_5$, of course. Also, I can realize $D_5$ as the subgroup of $W$ generated by five of the simple reflections. The problem is that I don't know how to get Sage to tell me which elements are in the conjugacy class (for example) labelled 6b in the E6 table---all I know about this class is that it consists of elements of order 6. For my purposes, I really need to know which elements these are, in matrix form, so that I can restrict the characters to $D_5$ and expand the result there in terms of irreducible $D_5$ characters.