Maximality of Laplacian Algebras, with Applications to Invariant Theory

TL;DR

This paper proves Laplacian algebras are maximal and applies this to solve classical invariant theory problems for real orthogonal representations of compact groups, including criteria for generating sets and new polarization techniques.

Contribution

It establishes the maximality of Laplacian algebras and introduces generalized polarizations that generate invariant algebras in certain representations.

Findings

Solved the Inverse Invariant Theory problem for finite groups.

Provided an if-and-only-if criterion for generating sets.

Introduced generalized polarizations that generate invariant algebras.

Abstract

We show Laplacian algebras are maximal, and give applications to the Classical Invariant Theory of real orthogonal representations of compact groups, including: The solution of the Inverse Invariant Theory problem for finite groups. An if-and-only-if criterion for when a separating set is a generating set. And the introduction of a class of generalized polarizations which, in a certain class of representations (including all representations of finite groups), always generates the algebra of invariants of their diagonal representations.