On stability of diagonal actions and tensor invariants

TL;DR

This paper investigates the connection between the existence of invariant tensors in tensor powers of modules for semisimple algebraic groups and the stability of diagonal group actions on algebraic varieties.

Contribution

It establishes new relations between invariant tensors and the stability of diagonal actions, advancing understanding in algebraic group actions and tensor invariants.

Findings

Existence of invariant tensors in tensor powers of modules for semisimple groups.

Relations between invariant tensors and stability of diagonal actions.

Conditions under which invariant tensors imply stability.

Abstract

For a connected simply connected semisimple algebraic group $G$ we prove existence of invariant tensors in tensor powers of rational $G$-modules and establish relations between existence of such invariant tensors and stability of diagonal actions of $G$ on affine algebraic varieties.