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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.

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