Splitting the difference: Computations of the Reynolds operator in classical invariant theory

TL;DR

This paper develops algorithms to compute the Reynolds operator for classical groups acting on polynomial rings, facilitating invariant theory calculations in algebraic geometry and representation theory.

Contribution

It introduces specific algorithms for computing the Reynolds operator for classical groups like GL, SL, O, and Sp with their standard and dual representations.

Findings

Algorithms successfully compute the Reynolds operator for classical groups.

Enhanced computational methods for invariant theory applications.

Provides explicit procedures for standard classical group actions.

Abstract

If $G$ is a linearly reductive group acting rationally on a polynomial ring $S$, then the inclusion $S^{G} \hookrightarrow S$ possesses a unique $G$-equivariant splitting, called the Reynolds operator. We describe algorithms for computing the Reynolds operator for the classical actions as in Weyl's book. The groups are the general linear group, the special linear group, the orthogonal group, and the symplectic group, with their classical representations: direct sums of copies of the standard representation and copies of the dual representation.