Symmetric invariants related to representations of exceptional simple groups

TL;DR

This paper classifies certain finite-dimensional rational representations of exceptional algebraic groups where the symmetric invariants of a related semi-direct product form a polynomial ring, advancing understanding of invariant theory in Lie algebra representations.

Contribution

It provides a classification of representations of exceptional groups with polynomial symmetric invariants in the semi-direct product setting.

Findings

Identified all such representations for exceptional algebraic groups.

Established conditions under which symmetric invariants form a polynomial ring.

Contributed to the broader understanding of invariant theory in Lie algebra representations.

Abstract

We classify the finite-dimensional rational representations $V$ of the exceptional algebraic groups $G$ with $\mathfrak g={\sf Lie}(G)$ such that the symmetric invariants of the semi-direct product $\mathfrak g\ltimes V$, where $V$ is an Abelian ideal, form a polynomial ring.