Supercharacter theory of an algebra group

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Suppose F is a finite field of size q, a prime power with underlying prime p. Suppose N is a nilpotent associative finite-dimensional algebra over F. Suppose, further, that G is the algebra group corresponding to N, i.e., G = 1 + N..

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

Superconjugacy classes

Note that G is a subgroup of the unitization N + F, hence acts on N + F by both left and right multiplication. The ideal N is invariant under both actions, so we get a left and a right G-action on N that commute with each other. Consider the double orbits of N under these actions, i.e., sets of the form GxG, x \in N. The corresponding sets 1 + GxG, which are subsets of G, are the superconjugacy classes.

Note that the conjugation action of G on N 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.