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arXiv · 1812.03606

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

G. I. Lehrer
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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.

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BibTeXRIS

G. I. Lehrer. 2018-12-10. A factorisation theorem for the coinvariant algebra of a unitary reflection group. https://arxiv.org/abs/1812.03606

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