# A factorisation theorem for the coinvariant algebra of a unitary reflection group

## TL;DR

This paper proves a new factorisation theorem for the coinvariant algebra of a complex unitary reflection group, generalising previous results and revealing a structured isomorphism involving harmonic polynomials and reflection subgroups.

## Contribution

It introduces a degree-preserving isomorphism between tensor products of harmonic polynomial spaces for a reflection group and its subgroup, extending prior work to complex unitary groups.

## Key findings

- Established a graded isomorphism for harmonic polynomial spaces
- Generalised a known result from real to complex reflection groups
- Enhanced understanding of the algebraic structure of coinvariant algebras

## Abstract

We prove the following theorem. Let $G$ be a finite group generated by unitary reflections in a complex Hermitian space $V=\mathbb{C}^\ell$ and let $G'$ be any reflection subgroup of $G$. Let $\mathcal{H}(G)$ be the space of $G$-harmonic polynomials on $V$. There is a degree preserving isomorphism $\xi:\mathcal{H}(G')\otimes\mathcal{H}(G)^{G'}\overset{\sim}{\longrightarrow}\mathcal{H}$ of graded $\mathcal{N}$-modules, where $\mathcal{N}:=N_{\rm{GL}(V)}(G)\cap N_{\rm{GL}(V)}(G')$ and $\mathcal{H}^{G'}$ is the space of $G'$-fixed points of $\mathcal{H}$. This generalises a result of Douglass and Dyer for parabolic subgroups of real reflection groups.