Braid moves in commutation classes of the symmetric group

作者：Schilling Anne^*; Thiery Nicolas M; White Graham; Williams Nathan

来源：European Journal of Combinatorics, 2017, 62: 15-34.

DOI：10.1016/j.ejc.2016.10.008

We prove that the expected number of braid moves in the commutation class of the reduced word (s(1)s(2)center dot center dot center dot sn-1)(s(1)s(2) center dot center dot center dot sn-2) center dot center dot center dot (s(1)s(2))(s(1)) for the long element in the symmetric group G(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.

• 出版日期2017-5