A Refined Count of Coxeter Element Reflection Factorizations

  • Elise delMas
  • Thomas Hameister
  • Victor Reiner
Keywords: Reflection group, Coxeter element, Factorization, Well-generated


For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the number

of reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.

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