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

Joshua P Swanson,Nolan R Wallach

doi:10.1016/j.jcta.2021.105474

Abstract

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.

Highlights

There has been a great deal of research activity in algebraic combinatorics studying diagonal actions of the symmetric group Sn on k sets of n commuting indeterminants and l sets of n anti-commuting indeterminants
Orellana–Zabrocki [13] describe the Sn-invariants of these polynomial rings combinatorially and summarize some of their history
The k = 2, l = 0 case has received a very large amount of attention through the study of the diagonal coinvariants Q[xn, yn]/Dn where Dn is the diagonal coinvariant ideal generated by all homogeneous Sninvariants of positive degree and xn is shorthand for x1, . . . , xn [9, 10]

Summary

Introduction

There has been a great deal of research activity in algebraic combinatorics studying diagonal actions of the symmetric group Sn on k sets of n commuting indeterminants and l sets of n anti-commuting indeterminants. Motivated by Zabrocki’s conjecture, the second author [28] gave a conjectural description of the harmonics of Q[xn, θn]/Jn where Jn is the super coinvariant ideal generated by homogeneous Sn-invariants of positive degree. This description further motivated Rhoades–Wilson [18] to recently construct another representation-theoretic model for the t = 0 specialization of the Delta conjecture arising from the leading terms of those harmonics. The classical coinvariant algebra of G is S(V ∗)/I∗ where I∗ is the ideal generated by all non-constant homogeneous G-invariants It is well-known that the top-degree component of S(V ∗)/I∗ is the image of S(V ∗)detV , which has motivated much of our work. A more explicit but less general version of many of these results that may be more palatable to algebraic combinatorialists can be found in [26]

Polynomial differential forms

Harmonics and semi-invariant bases

Coinvariant harmonics and semi-invariants