Theodosis Douvropoulos, Institut de Recherche en Informatique Fontamentale: Coxeter numbers: From fake degree palindromicity to the enumeration of reflection factorizations of regular elements

Referent: Theodosis Douvropoulos, Institut de Recherche en Informatique Fontamentale: Coxeter numbers: From fake degree palindromicity to the enumeration of reflection factorizations of regular elements

Zeit:Mittwoch, 21.11.2018, 17:15 h

Ort:Raum 48-436

Abstract:

A famous theorem of Cayley states that there are n^^n-2 vertex-labeled trees on n vertices. A different interpretation of this number, due to Hurwitz, is that it counts smallest length factorizations of the Coxeter element (12...n) of S_ⁿ in transpositions. As with many fascinating theorems for the Symmetric group, this statement is the shadow of a result that holds for all reflection groups. Bessis has shown after case-by-case calculations that: The number of smallest length reflection factorizations of a Coxeter element c of W is equal to hⁿ*n!/|W|, where h is the order of the Coxeter element and n the rank of W. We will present in this talk a uniform argument for this statement (that however relies on the BMR-freeness theorem) and partial generalizations of it that hold for arbitrary regular elements and arbitrary length factorizations. A classical approach due to Frobenius translates the above problem into a series of character evaluations. In the absence of a uniform construction of the characters χ of complex reflection groups, our method first groups the characters together with respect to an invariant called the Coxeter number $cχ$. It proceeds by making use of a Galois action on the corresponding Hecke algebra characters which was first considered by Malle to prove a palindromicity phenomenon on the fake degrees of W.