A supercharacter analogue for normality

Diaconis and Isaacs define in [8] (Diaconis and Isaacs, 2008) a supercharacter theory for algebra groups over a finite field by constructing certain unions of conjugacy classes called superclasses and certain reducible characters called supercharacters. This work investigates the properties of algebra subgroups H⊂G which are unions of some set of the superclasses of G; we call such subgroups supernormal. After giving a few useful equivalent formulations of this definition, we show that products of supernormal subgroups are supernormal and that all normal pattern subgroups are supernormal. We then classify the set of supernormal subgroups of Un(q), the group of unipotent upper triangular matrices over the finite field Fq, and provide a formula for the number of such subgroups when q is prime. Following this, we give supercharacter analogues for Cliffordʼs theorem and Mackeyʼs “method of little groups.” Specifically, we show that a supercharacter restricted to a supernormal subgroup decomposes as a sum of supercharacters with the same degree and multiplicity. We then describe how the supercharacters of an algebra group of the form Un=Uh⋉Ua, where Ua is supernormal and a2=0, are parametrized by Uh-orbits of the supercharacters of Ua and the supercharacters of the stabilizer subgroups of these orbits.