# Cyclic sieving for reduced reflection factorizations of the Coxeter element

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.