Multiplicative Invariants and Semigroup Algebras

TL;DR

This paper studies when algebras of multiplicative invariants, arising from finite group actions on lattices, are semigroup algebras, providing explicit conditions and counterexamples.

Contribution

It offers an explicit characterization of when multiplicative invariants form semigroup algebras, including a proof for reflection groups and counterexamples for fixed point free actions.

Findings

Multiplicative invariants of finite reflection groups are semigroup algebras.

Fixed point free actions do not produce semigroup algebras.

G having odd prime order implies invariants are not semigroup algebras.

Abstract

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We investigate when algebras of multiplicative invariants are semigroup algebras. In particular, we present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are indeed semigroup algebras. On the other hand, multiplicative invariants arising from fixed point free actions are shown to never be semigroup algebras. In particular, this holds whenever G has odd prime order.