James Taylor

Character Theory for Semilinear Representations

Semilinear representations are a natural generalisation of linear representations, where now there is also an action of the group on the coefficient field itself. We describe recent results which allow one to describe such categories, by reducing the problem to determining certain associated natural numbers known as semilinear Schur indices.

In this talk, we give an overview of this theory, and discuss recent work on determining these semilinear Schur indices in two cases of interest: when the group is Z, and when the action of the group on the field is in the simplest non-trivial case (when the fixed field has index two). From the first, we obtain an explicit description of the category of isocrystals over a finite field, and from the second, a generalisation of Freeman Dyson's famous "Tenfold Way".