Modular Invariants for Some Finite Pseudo-Reflection Groups

TL;DR

This paper determines the structure of modular invariants for certain finite pseudo-reflection groups within the general linear group over finite fields, focusing on subgroups related to parabolic and Weyl groups.

Contribution

It provides a detailed analysis of modular invariants for pseudo-reflection subgroups of GL_n(q), extending understanding to groups associated with Cartan type Lie algebras.

Findings

Identified the invariants for specific pseudo-reflection groups

Extended known results to groups related to Cartan type Lie algebras

Provided explicit descriptions of invariant rings in modular settings

Abstract

In this paper, we determine the modular invariants of finite modular pseudo-reflection subgroups of the finite general linear group $ \text{GL}_n(q) $ acting on the tensor product of the symmetric algebra $ S^{\bullet}(V) $ and the exterior algebra $ \wedge^{\bullet}(V) $ of the natural $ \text{GL}_n(q) $-module $ V $. We are particularly interested in the case where $ G $ is a subgroup of the parabolic subgroups of $ \text{GL}_n(q) $ which is a generalization of Weyl group of Cartan type Lie algebra.