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arXiv · 2001.06076

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

Joshua P. Swanson·Nolan R. Wallach
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Abstract

We give a type-independent construction of an explicit basis for the semi-invariant harmonic differential forms of an arbitrary pseudo-reflection group in characteristic zero. Our "top-down" approach uses the methods of Cartan's exterior calculus and is in some sense dual to related work of Solomon, Orlik--Solomon, and Shepler describing (semi-)invariant differential forms. We apply our results to a recent conjecture of Zabrocki which provides a representation theoretic-model for the Delta conjecture of Haglund--Remmel--Wilson in terms of a certain non-commutative coinvariant algebra for the symmetric group. In particular, we verify the alternating component of a specialization of Zabrocki's conjecture.

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Joshua P. Swanson, Nolan R. Wallach. 2020-01-16. Harmonic differential forms for pseudo-reflection groups I. Semi-invariants. https://arxiv.org/abs/2001.06076

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