On the Hurwitz action in finite Coxeter groups

We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a factorization into a product of reflections that generate a parabolic subgroup. We give an unusual definition of a parabolic subgroup that we show to be equivalent to the classical one for finite Coxeter groups.

Balanced labellings of affine permutations

International audience We study the $\textit{diagrams}$ of affine permutations and their $\textit{balanced}$ labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the $\textit{column strict}$ balanced labellings is the affine Stanley symmetric function defined by Lam and we give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we give a necessary and sufficient condition for a diagram to be an affine permutation diagram. Finally, we conjecture that if two affine permutations are $\textit{diagram equivalent}$ then their affine Stanley symmetric functions coincide.

Reduced Decompositions of Matchings

We give a characterization of matchings in terms of the canonical reduced decompositions. As an application, the canonical reduced decompositions of $12312$-avoiding matchings are obtained. Based on such decompositions, we find a bijection between $12312$-avoiding matchings and ternary paths.

An algorithm which generates linear extensions for a generalized Young diagram with uniform probability

International audience The purpose of this paper is to present an algorithm which generates linear extensions for a generalized Young diagram, in the sense of D. Peterson and R. A. Proctor, with uniform probability. This gives a proof of a D. Peterson's hook formula for the number of reduced decompositions of a given minuscule elements. \par Le but de ce papier est présenter un algorithme qui produit des extensions linéaires pour un Young diagramme généralisé dans le sens de D. Peterson et R. A. Proctor, avec probabilité constante. Cela donne une preuve de la hook formule d'un D. Peterson pour le nombre de décompositions réduites d'un éléments minuscules donné.

Minimal Factorizations of Permutations into Star Transpositions

International audience We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form $(1 i)$. Our result generalizes earlier work of Pak ($\textit{Reduced decompositions of permutations in terms of star transpositions, generalized catalan numbers and k-ary trees}$, Discrete Math. $\textbf{204}$:329―335, 1999) in which substantial restrictions were placed on the permutation being factored. Nous présentons une expression compacte pour le nombre de factorisations minimales d'une permutation arbitraire de transposition de la forme $(1 i)$. Ce résultat généralise le travail passé de Pak ($\textit{Reduced decompositions of permutations in terms of star transpositions, generalized catalan numbers and k-ary trees}$, Discrete Math. $\textbf{204}$:329―335, 1999) dans lequel des restrictions substantielles sont imposées sur la permutation étant factorisée.