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Braid moves in commutation classes of the symmetric group. (English) Zbl 1358.05308
Summary: 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 [ibid. 26, No. 6, 1019–1021 (2005; Zbl 1071.20003)], 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 Viennot’s theory of heaps and variants of the promotion operator [G. X. Viennot, Lect. Notes Math. 1234, 321–350 (1986; Zbl 0618.05008)]. 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.

05E10 Combinatorial aspects of representation theory
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20B30 Symmetric groups
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