Harmonic differential forms for pseudo-reflection groups I. Semi-invariants

We provide a type-independent construction of an explicit basis for the semi-invariant harmonic differential forms of an arbitrary pseudo-reflection group in characteristic zero. Equivalently, we completely describe the structure of the chi-isotypic components of the corresponding super coinvariant algebras in one commuting and one anti-commuting set of variables, for all linear characters chi. In type A, we verify a specialization of a conjecture of Zabrocki [37] which provides a representation-theoretic model for the Delta conjecture of Haglund-Remmel-Wilson [10]. Our top-down approach uses the methods of Cartan's exterior calculus and is in some sense dual to related work of Solomon [29], Orlik-Solomon [21], and Shepler [27,28] describing (semi-)invariant differential forms. (C) 2021 The Author(s). Published by Elsevier Inc.

Automated summary

The paper presents a type-independent construction for explicit basis of semi-invariant harmonic differential forms of pseudo-reflection groups in characteristic zero. It completely describes the structure of chi-isotypic components of corresponding super coinvariant algebras, and verifies a specialization of a conjecture, providing a representation-theoretic model for the Delta conjecture. The top-down approach using Cartan's exterior calculus is dual to related work describing invariant differential forms.