Reflection Groups and Differential Forms

TL;DR

This paper investigates invariant differential forms under finite reflection groups, establishing a freeness criterion and exploring algebraic structures, with extensions to relative invariants, advancing understanding in invariant theory and algebraic geometry.

Contribution

It introduces an analogue of Saito's freeness criterion for invariant differential 1-forms and analyzes the algebraic structure of invariant forms under reflection groups.

Findings

Proved an analogue of Saito's freeness criterion for invariant differential 1-forms.

Showed that twisted wedging can give invariant forms a free exterior algebra structure.

Extended results to relative invariants with respect to linear characters.

Abstract

We study differential forms invariant under a finite reflection group over a field of arbitrary characteristic. In particular, we prove an analogue of Saito's freeness criterion for invariant differential 1-forms. We also discuss how twisted wedging endows the invariant forms with the structure of a free exterior algebra in certain cases. Some of the results are extended to the case of relative invariants with respect to a linear character.