Modular invariants of finite gluing groups

TL;DR

This paper explores modular invariants of finite gluing groups using a novel construction, providing explicit generators and analyzing invariant fields for various classical groups and their subgroups.

Contribution

It introduces parabolic and thin gluing methods to compute invariants and faithful representations, advancing understanding of modular invariants in classical groups.

Findings

Generated explicit invariant rings for maximal parabolic subgroups.

Computed invariant fields of fractions for several representations.

Determined minimal dimensions for faithful representations of specific semidirect products.

Abstract

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.