We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow $p$-subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this construction to compute the invariant field of fractions for a range of representations. We use thin gluing to construct faithful representations of semidirect products and to determine the minimum dimension of a faithful representation of the semidirect product of a cyclic $p$-group acting on an elementary abelian $p$-group.

中文翻译：

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有限粘合群的模不变量

我们使用 Jia Huang 引入的粘合构造来探索一系列模块化表示的不变量环。我们为有限辛群的最大抛物子群的不变量环及其公共 Sylow $p$-subgroup 构造生成集。我们还研究了奇异有限经典群的不变量。我们引入抛物线粘合并使用这种构造来计算一系列表示的分数的不变域。我们使用薄粘合来构造半直接积的忠实表示，并确定作用于基本阿贝尔 $p$ 群的循环 $p$-群的半直接积的忠实表示的最小维数。