# Braid moves in commutation classes of the symmetric group

1 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)
LRI - Laboratoire de Recherche en Informatique
Abstract : We prove that the expected number of braid moves in the commutation class of the reduced word $(s_1 s_2 \cdots s_{n-1})(s_1 s_2 \cdots s_{n-2}) \cdots (s_1 s_2)(s_1)$ for the long element in the symmetric group $\mathfrak{S}_n$ is one. This is a variant of a similar result by V. Reiner, who proved that the expected number of braid moves in a random reduced word for the long element is one. The proof is bijective and uses X. Viennot's theory of heaps and variants of the promotion operator. In addition, we provide a refinement of this result on orbits under the action of even and odd promotion operators. This gives an example of a homomesy for a nonabelian (dihedral) group that is not induced by an abelian subgroup. Our techniques extend to more general posets and to other statistics.
Type de document :
Article dans une revue
European Journal of Combinatorics, Elsevier, 2017, 62, pp.15 - 34. 〈10.1016/j.ejc.2016.10.008〉
Domaine :
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https://hal.archives-ouvertes.fr/hal-01611482
Contributeur : Nicolas M. Thiéry <>
Soumis le : jeudi 5 octobre 2017 - 22:42:32
Dernière modification le : mardi 24 avril 2018 - 13:34:35

### Citation

Thiéry Nicolas, Anne Schilling, Graham White, Nathan Williams. Braid moves in commutation classes of the symmetric group. European Journal of Combinatorics, Elsevier, 2017, 62, pp.15 - 34. 〈10.1016/j.ejc.2016.10.008〉. 〈hal-01611482〉

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