# Cyclic sieving for reduced reflection factorizations of the Coxeter element

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Time: 11:00 -- Location: Salle Philippe Flajolet du LIX

Given a factorization $$t_1\cdots t_n=c$$ of some element $$c$$ in a group, there are various natural cyclic operations we can apply on it; one of them is given by $$\Psi:(t_1,\cdots,t_n)\rightarrow (c\ t_n\ c^{-1},t_1,\cdots, t_{n-1})$$. A common question is then to enumerate factorizations with a certain degree of symmetry, that is, those for which $$\Psi^k$$ 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 $$(nh/k)^{\text{th}}$$ 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 $$d_i$$'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 $$\Psi$$. This is, in fact, the (combinatorial) heart of the argument.

Category: seminars
Tags: Combi seminar combinatorics