$U_n(q)$的行闭子群的余伴随轨道
Coadjoint orbits of Row Closed Subgroups of $U_n(q)$
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中文总结 AI 辅助
本文对有限域上单位上三角矩阵群的行闭子群的余伴随轨道进行分类，确定轨道并识别同构及正交超特征对应的轨道模，列闭子群分类由镜像映射导出。
中文摘要 AI 辅助
设$n$为自然数，$q$为素数幂，$U_n(q)$表示有限域$\boldsymbol{F}_q$上的$n\times n$阶单位上三角矩阵群。$U_n(q)$的行闭子群与列闭子群是特殊的模式子群，分别通过删除整行或整列（对角元除外）得到。André-Yan超特征理论中$U$的超特征由$U$在其李代数的特征群上的单项作用产生的轨道模实现。本文对行闭子群$U$对应的轨道进行分类，即确定所有轨道，并识别哪些关联轨道模同构，哪些实现正交超特征。列闭子群的分类可通过沿反对角线反射矩阵的镜像映射得到。
英文摘要
Let $n$ be a natural number, let $q$ be a prime power, and let $U_n(q)$ denote the group of unitriangular $n\times n$ matrices over the finite field $\mathbb{F}_q$ with $q$ elements. Row closed and column closed subgroups $U$ of $U_n(q)$ are special pattern subgroups obtained by deleting entire rows or, respectively, entire columns (apart from the diagonal entries). The supercharacters of the André-Yan supercharacter theory of $U$ are afforded by the orbit modules arising from a monomial action of $U$ on the character group of the Lie algebra of $U$. We classify the corresponding orbits for row closed subgroups $U$, that is, we determine all orbits and identify which of the associated orbit modules are isomorphic and which afford orthogonal supercharacters. The classification for column closed subgroups follows from the mirror map, which reflects matrices across the antidiagonal.
