# Supercharacter theory of an algebra group

## Definition

Suppose is a finite field of size , a prime power with underlying prime . Suppose is a nilpotent associative finite-dimensional algebra over . Suppose, further, that is the algebra group corresponding to , i.e., ..

The **supercharacter theory** of is a supercharacter theory defined as follows.

### Superconjugacy classes

Note that is a subgroup of the unitization , hence acts on by both left and right multiplication. The ideal is invariant under both actions, so we get a left and a right -action on that commute with each other. Consider the double orbits of under these actions, i.e., sets of the form . The corresponding sets , which are subsets of , are the superconjugacy classes.

Note that the conjugation action of on corresponds to its conjugation action on itself, and the orbits for this conjugation action are contained inside the double orbits, so the superconjugacy classes are unions of conjugacy classes.