Multiplicative Invariants of Root Lattices

TL;DR

This paper characterizes the multiplicative invariants of root lattices under Weyl group actions, identifying fundamental invariants, class groups, and algebra presentations for all irreducible root systems.

Contribution

It provides a comprehensive description of the invariant algebras, including fundamental invariants, class groups, and explicit algebra presentations, advancing understanding of root lattice symmetries.

Findings

Fundamental invariants for all irreducible root systems identified.

Class groups of the invariant algebras computed.

Presentations and Hironaka decompositions provided where applicable.

Abstract

We describe the multiplicative invariant algebras of the root lattices of all irreducible root systems under the action of the Weyl group. In each case, a finite system of fundamental invariants is determined and the class group of the invariant algebra is calculated. In some cases, a presentation and a Hironaka decomposition of the invariant algebra is given.