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

2018/12/10 by G Lehrer, Lehrer, G. I.
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1812.03606

Abstract

We prove the following theorem. Let G be a finite group generated by unitary reflections in a complex Hermitian space V=ℂ^ℓ and let G' be any reflection subgroup of G. Let H(G) be the space of G-harmonic polynomials on V. There is a degree preserving isomorphism ξ:H(G')\otimesH(G)^G'\overset∼\longrightarrowH of graded N-modules, where N:=N_\rmGL(V)(G)∩ N_\rmGL(V)(G') and H^G' is the space of G'-fixed points of H. This generalises a result of Douglass and Dyer for parabolic subgroups of real reflection groups.

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