Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups

TL;DR

This paper characterizes strong Lefschetz elements in the coinvariant rings of finite Coxeter groups, establishing conditions based on reflection invariance, and extends results to invariant subrings of Weyl groups.

Contribution

It provides a complete characterization of strong Lefschetz elements in coinvariant rings of finite Coxeter groups, excluding type H4, and describes conditions for invariant subrings of Weyl groups.

Findings

Homogeneous degree-one elements are strong Lefschetz iff not fixed by reflections.

Necessary and sufficient conditions for strong Lefschetz elements in invariant subrings.

Results exclude Coxeter group type H4 due to specific structural differences.

Abstract

For the coinvariant rings of finite Coxeter groups of types other than H$_4$, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.