Relative equivariants under compact Lie groups

TL;DR

This paper characterizes polynomial mappings that commute with linear actions of relative symmetry groups, providing algorithms to compute generators for relative equivariant modules based on invariant theory.

Contribution

It extends previous invariant results to relative equivariants, offering a method to compute generators from subgroup invariants and equivariants.

Findings

Derived the general form of polynomial relative equivariants.

Developed an algorithm to compute generators for relative equivariant modules.

Provided a method to obtain equivariant generators from subgroup data.

Abstract

In this work we obtain the general form of polynomial mappings that commute with a linear action of a relative symmetry group. The aim is to give results for relative equivariant polynomials that correspond to the results for relative invariants obtained in a previous paper [P.H. Baptistelli, M. Manoel (2013) Invariants and relative invariants under compact Lie groups, J. Pure Appl. Algebra 217, 2213{2220]. We present an algorithm to compute generators for relative equivariant submodules from the invariant theory applied to the subgroup formed only by the symmetries. The same method provides, as a particular case, generators for equivariants under the whole group from the knowledge of equivariant generators by a smaller subgroup, which is normal of finite index.