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 χ-isotypic components of the corresponding super coinvariant al...

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Bibliographic Details
Published inJournal of combinatorial theory. Series A Vol. 182; p. 105474
Main Authors Swanson, Joshua P., Wallach, Nolan R.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.08.2021
Subjects
Coinvariant algebras
Harmonics
Invariant theory
Pseudo-reflection groups
Semi-invariants
Pseudo-reflection groups
Semi-invariants
Harmonics
Invariant theory
Coinvariant algebras
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Summary: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 χ-isotypic components of the corresponding super coinvariant algebras in one commuting and one anti-commuting set of variables, for all linear characters χ. 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.
ISSN:0097-3165
1096-0899
DOI:10.1016/j.jcta.2021.105474