Linear preservers and representations with a 1-dimensional ring of invariants

TL;DR

This paper characterizes the group of linear transformations that preserve specific polynomial functions on vector spaces, utilizing the theory of semisimple algebraic groups to identify these symmetry groups.

Contribution

It provides a classification of linear preservers for polynomial functions on vector spaces, connecting invariant theory with algebraic group actions.

Findings

Identifies the linear groups preserving given polynomial functions.

Uses semisimple algebraic group theory to analyze invariants.

Provides explicit descriptions for various pairs (V,f).

Abstract

We determine the group of linear transformations on a vector space $V$ that preserve a polynomial function $f$ on $V$ for several interesting pairs $(V,f)$, using the theory of semisimple algebraic groups.