A Supercharacter Analogue for Normality

Abstract

Diaconis and Isaacs define 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 $H\subset G$ 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 $U_n(q)$, the group of unipotent upper triangular matrices over the finite field $\FF_q$, and provide a formula for the number of such subgroups when $q$ is prime. Following this, we give supercharacter analogues for Clifford's theorem and Mackey's "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 $U_\fkn = U_\fkh \ltimes U_\fka$, where $U_\fka$ is supernormal and $\fka^2=0$, are parametrized by $U_\fkh$-orbits of the supercharacters of $U_\fka$ and the supercharacters of the stabilizer subgroups of these orbits.

35 pages

References in corpus (4)

• Superinduction for pattern groups
• Irreducible Characters of Finite Algebra Groups
• Construction of supercharacter theories of finite groups
• Nonzero coefficients in restrictions and tensor products of supercharacters of $U_n(q)$

Cited by in corpus (10)

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• Superclasses and supercharacters of normal pattern subgroups of the unipotent upper triangular matrix group
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• Toward a Classification of the Supercharacter Theories of $C_p\times C_p$
• On supercharacter theoretic generalizations of monomial groups and Artin's conjecture
• Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes
• On the monomial algebra associated to the monomial characters of a finite group
• A unipotent realization of the chromatic quasisymmetric function