正规性的一个超特征标
A Supercharacter Analogue for Normality
摘要：
Diaconis和Isaacs定义了一个有限域上的代数群的超特征标理论，通过构造某些共轭类（称为超类）和某些可约特征标（称为超特征标）的并集。本文研究了代数子群$H\subset G$的性质，它是$G$的超类的集合的并集，我们称这种子群为超正规子群。在给出这个定义的几个有用的等价公式后，我们证明了超正规子群的乘积是超正规的，并且所有正规模式子群都是超正规的。然后对有限域$\FF_q$上的幂单上三角形矩阵群$U_n(q)$的超正规子群集进行分类，并给出了当$q$为素数时，超正规子群个数的计算公式.在此基础上，我们给出了Clifford定理和Mackey的“小群方法”的超特征标类似物。“具体来说，我们证明了一个限制在一个超正规子群的超字符可以分解为具有相同次数和多重性的超字符之和。然后，我们描述了如何的超字符的代数群的形式$U_\fkn = U_\fkh \ltimes U_\fka$，其中$U_\fka$是supernormal和$\fka^2=0$，参数化的$U_\fkh$ -轨道的超字符的$U_\fka$和超字符的稳定子群这些轨道。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
