# Exposé par Theo Douvropoulos: «Cyclic sieving for reduced reflection factorizations of the Coxeter element»

Speaker: Theo Douvropoulos
Location: Salle Philippe Flajolet
Date: Wed, 20 Jun 2018, 11:00-12:00

La prochaine séance du séminaire Combi du Plateau de Saclay aura lieu ce mercredi 20 juin à 11h dans la salle Philippe Flajolet du LIX. Nous aurons le plaisir d'écouter Theo Douvropoulos (IRIF, Univ. Paris 7) nous parler de «Cyclic sieving for reduced reflection factorizations of the Coxeter element». Le résumé est disponible ci-dessous.

Le programme du séminaire est disponible ici : https://galac.lri.fr/pages/combi-seminar.html

Résumé: Given a factorization t1tn = c of some element c in a group, there are various natural cyclic operations we can apply on it; one of them is given by:

Ψ : (t1, ⋯, tn)→(c tn c−1, t1, ⋯, tn − 1).

A common question is then to enumerate factorizations with a certain degree of symmetry, that is, those for which Ψk is the identity, for a given k. In this talk we present the case of minimal length reflection factorizations of the Coxeter element c of a (well-generated, complex) reflection group W. The answer is a satisfying cyclic-sieving phenomenon; namely the enumeration is given by evaluating the following polynomial (in q) on a (nh/k)th root of unity:

$$X(q):=\prod_{i=1}^n\dfrac{[hi]_q}{[d_i]_q},\quad\quad\text{ where }[n]_q:=\dfrac{1-q^n}{1-q}.$$

Here, h is the Coxeter number (the order of c) and the di's are the fundamental invariants of W. The proof is based on David Bessis' pioneering work, which gave a geometric interpretation for such factorizations. We will only sketch the necessary background material and focus more on a topological braid-theoretic representation of cyclic operations such as Ψ. This is, in fact, the (combinatorial) heart of the argument.