Equivariant Euler characteristics of discriminants of reflection groups

2002/01/16 by Graham Denham, Denham, Graham, Nicole Lemire +1
Mathematics · #20F55 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CO #math.GR #msc:20F55

paper · pdf · doi:10.48550/arxiv.math/0201150

18 pages

arxiv created 2002/01/16 · openalex publication_date 2002/01/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a finite, complex reflection group and f its discriminant polynomial. The fibers of f admit commuting actions of G and a cyclic group. The virtual G× C_m character given by the Euler characteristic of the fiber is a refinement of the zeta function of the geometric monodromy, calculated in a paper of Denef and Loeser. We compute the virtual character explicitly, in terms of the poset of normalizers of centralizers of regular elements of G, and of the subspace arrangement given by proper eigenspaces of elements of G. As a consequence, we compute orbifold Euler characteristics and find some new "case-free" information about the discriminant.

Related

• Reflection groups acting on their hyperplanes
• Invariant differential derivations for reflection groups in positive characteristic
• On equivariant versions of higher order orbifold Euler characteristics
• Orbifold Euler characteristics and the number of commuting n-tuples in symmetric groups
• Mirror Principle I