On some properties of LS algebras

TL;DR

This paper explores the algebraic properties of LS algebras, showing they are invariant rings of certain finite abelian groups, Gorenstein, and Koszul, with implications for toric varieties.

Contribution

It establishes that LS algebras are invariant rings of finite abelian groups without pseudo-reflections, Gorenstein, and Koszul, advancing understanding of their algebraic structure.

Findings

LS algebra is the invariant ring of a finite abelian group without pseudo-reflections.

LS algebra is Gorenstein under certain conditions.

All LS algebras are Koszul.

Abstract

The discrete LS algebra over a totally ordered set is the homogeneous coordinate ring of an irreducible projective (normal) toric variety. We prove that this algebra is the ring of invariants of a finite abelian group containing no pseudo-reflection acting on a polynomial ring. This is used to study the Gorenstein property for LS algebras. Further we show that any LS algebra is Koszul.