Normal Reflection Subgroups of Complex Reflection Groups
Carlos E. Arreche, Nathan F. Williams
TL;DR
This work studies normal reflection subgroups $N$ of complex reflection groups $G$ and provides a refined Os–Orlik–Solomon framework that accounts for the quotient $H=G/N$. Central results include a numerology relation $e_i^N(M)+e_i^G(U_M^N)=e_i^G(M)$ and a twisted product formula $\\sum_{g\in G}(\\prod_{\\lambda_i(g)\\neq 1} \\frac{1-\\lambda_i(g)^\\sigma}{1-\\lambda_i(g)}) q^{\\mathrm{fix}}_V(g) t^{\\mathrm{fix}}_E(g)=\\prod_{i=1}^r (qt+e_i^N(V^\\sigma) t + e_i^G(U^N_\\sigma))$, unifying OS theory with Galois twists. The authors develop and relate OS-spaces ${U^N_\\sigma}$, harmonic coinvariants, and amenable representations to derive two independent derivations of the main product formula, generalizing prior case-by-case results. They also classify normal reflection subgroups and provide thorough examples, including cyclic groups, the infinite family $G(ab,b,r)$, and a notable exceptional case, illustrating the broader applicability to invariant theory and hyperplane-complement cohomology. The framework connects invariant theory, representation theory, and topological aspects of hyperplane arrangements, offering a uniform approach to questions previously treated case-by-case.
Abstract
We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalized exponents. Our refinement gives a uniform proof and generalization of a recent theorem of the second author.