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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. Viennots 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.
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Using the standard Coxeter presentation for the symmetric group $S_n$, two reduced expressions for the same group element are said to be commutation equivalent if we can obtain one expression from the other by applying a finite sequence of commutations. The resulting equivalence classes of reduced expressions are called commutation classes. How many commutation classes are there for the longest element in $S_n$?
In this article we describe the summit sets in B_3, the smallest element in a summit set and we compute the Hilbert series corresponding to conjugacy classes.The results will be related to Birman-Menesco classification of knots with braid index three or less than three.
We construct a 2-representation categorifying the symmetric Howe representation of $mathfrak{gl}_m$ using a deformation of an algebra introduced by Webster. As a consequence, we obtain a categorical braid group action taking values in a homotopy category.
We study maps on the set of permutations of n generated by the Renyi-Foata map intertwined with other dihedral symmetries (of a permutation considered as a 0-1 matrix). Iterating these maps leads to dynamical systems that in some cases exhibit interesting orbit structures, e.g., every orbit size being a power of two, and homomesic statistics (ones which have the same average over each orbit). In particular, the number of fixed points (aka 1-cycles) of a permutation appears to be homomesic with respect to three of these maps, even in one case where the orbit structures are far from nice. For the most interesting such Foatic action, we give a heap analysis and recursive structure that allows us to prove the fixed-point homomesy and orbit properties, but two other cases remain conjectural.
A family of permutations $mathcal{F} subset S_{n}$ is said to be $t$-intersecting if any two permutations in $mathcal{F}$ agree on at least $t$ points. It is said to be $(t-1)$-intersection-free if no two permutations in $mathcal{F}$ agree on exactly $t-1$ points. If $S,T subset {1,2,ldots,n}$ with $|S|=|T|$, and $pi: S to T$ is a bijection, the $pi$-star in $S_n$ is the family of all permutations in $S_n$ that agree with $pi$ on all of $S$. An $s$-star is a $pi$-star such that $pi$ is a bijection between sets of size $s$. Friedgut and Pilpel, and independently the first author, showed that if $mathcal{F} subset S_n$ is $t$-intersecting, and $n$ is sufficiently large depending on $t$, then $|mathcal{F}| leq (n-t)!$; this proved a conjecture of Deza and Frankl from 1977. Equality holds only if $mathcal{F}$ is a $t$-star. In this paper, we give a more `robust proof of a strengthening of the Deza-Frankl conjecture, namely that if $n$ is sufficiently large depending on $t$, and $mathcal{F} subset S_n$ is $(t-1)$-intersection-free, then $|mathcal{F} leq (n-t)!$, with equality only if $mathcal{F}$ is a $t$-star. The main ingredient of our proof is a `junta approximation result, namely, that any $(t-1)$-intersection-free family of permutations is essentially contained in a $t$-intersecting {em junta} (a `junta being a union of a bounded number of $O(1)$-stars). The proof of our junta approximation result relies, in turn, on a weak regularity lemma for families of permutations, a combinatorial argument that `bootstraps a weak notion of pseudorandomness into a stronger one, and finally a spectral argument for pairs of highly-pseudorandom fractional families. Our proof employs four different notions of pseudorandomness, three being combinatorial in nature, and one being algebraic.
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