Abstract
AbstractWe 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 generalised exponents. Our refinement gives a uniform proof and generalisation of a recent theorem of the second author.
Highlights
Chevalley presented these ei for the exceptional simple Lie algebras in his 1950 address at the International Congress of Mathematicians [8], and Coxeter recognised them from previous work with real reflection groups [10]
This observation has led to deep relationships between the cohomology of G and the invariant theory of the corresponding Weyl group W = NG(T )/T, where T is a maximal torus in G [18, 17] – notably, H∗(G) ≃ (H∗(G/T ) × H∗(T ))W ≃ S(V ∗)/IW+ ⊗
Shephard and Todd verified case by case that the same sum still factors when W is replaced by a finite complex reflection group G ⊂ GL(V ) acting by reflections on a complex vector space V of dimension r [19, Theorem 5.3]
Summary
Chevalley presented these ei for the exceptional simple Lie algebras in his 1950 address at the International Congress of Mathematicians [8], and Coxeter recognised them from previous work with real reflection groups [10]. This observation has led to deep relationships between the cohomology of G and the invariant theory of the corresponding Weyl group W = NG(T )/T , where T is a maximal torus in G [18, 17] – notably, H∗(G) ≃ (H∗(G/T ) × H∗(T ))W ≃ S(V ∗)/IW+ ⊗. We refer the reader to the wonderful survey [3]
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