Invariant derivations and differential forms for reflection groups
Abstract
Classical invariant theory of a complex reflection group highlights three beautiful structures: -- the -invariant polynomials constitute a polynomial algebra, over which -- the -invariant differential forms with polynomial coefficients constitute an exterior algebra, and -- the relative invariants of any -representation constitute a free module. When is a duality (or well-generated) group, we give an explicit description of the isotypic component within the differential forms of the irreducible reflection representation. This resolves a conjecture of Armstrong, Rhoades and the first author, and relates to Lie-theoretic conjectures and results of Bazlov, Broer, Joseph, Reeder, and Stembridge, and also Deconcini, Papi, and Procesi. We establish this result by examining the space of -invariant differential derivations; these are derivations whose coefficients are not just polynomials, but differential forms with polynomial coefficients. For every complex reflection group , we show that the space of invariant differential derivations is finitely generated as a module over the invariant differential forms by the basic derivations together with their exterior derivatives. When is a duality group, we show that the space of invariant differential derivations is free as a module over the exterior subalgebra of -invariant forms generated by all but the top-degree exterior generator. (The basic invariant of highest degree is omitted.) Our arguments for duality groups are case-free, i.e., they do not rely on any reflection group classification.
Cite
@article{arxiv.1612.01031,
title = {Invariant derivations and differential forms for reflection groups},
author = {Victor Reiner and Anne V. Shepler},
journal= {arXiv preprint arXiv:1612.01031},
year = {2019}
}
Comments
Minor revisions; version to appear in Proc. Lond. Math. Soc