Let U(q) denote the set of infinite nilpotent matrices over the finite field with q elements, called infinite unitriangular group. It is a wild group (not type I), thus the irreducible representation do not admit any reasonable parametrization.
Nevertheless the set of all characters (positive definite, normalized class functions) characterize up to quasi-equivalence
representations of type I and II, therefore, they are our main object of study.
The goal of this talk is to define what are Supercharacter-Theories for U(q) and describe one in particular. This description relies on set partitions and some combinatorial relations.