Invariants and relative invariants under compact Lie groups

TL;DR

This paper develops algebraic methods to analyze polynomial relative invariants under compact Lie group actions, focusing on subgroup normality and constructing invariant generators using Hilbert bases and Reynolds operators.

Contribution

It introduces a systematic approach to compute invariants and relative invariants for compact Lie groups with normal subgroups, expanding invariant theory techniques.

Findings

Decomposition of H-invariants into submodules over the ring of G-invariants

Construction of G-invariants from H-invariants using Hilbert bases

Illustrative examples demonstrating the methods

Abstract

This paper presents algebraic methods for the study of polynomial relative invariants, when the group G formed by the symmetries and relative symmetries is a compact Lie group. We deal with the case when the subgroup H of symmetries is normal in G with index m, m greater or equal to 2. For this, we develop the invariant theory of compact Lie groups acting on complex vector spaces. This is the starting point for the study of relative invariants and the computation of their generators. We first obtain the space of the invariants under the subgroup $H$ of $\Gamma$ as a direct sum of $m$ submodules over the ring of invariants under the whole group. Then, based on this decomposition, we construct a Hilbert basis of the ring of G-invariants from a Hilbert basis of the ring of H-invariants. In both results the knowledge of the relative Reynolds operators defined on H-invariants is shown to be an essential tool to obtain the invariants under the whole group. The theory is illustrated with some examples.