Abstract
Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.
Highlights
In [5], the second-named author defines for each affine Weyl group an affine variety whose integral points are in bijection with the group
In the case of the affine symmetric group Sn, it turns out that the irreducible components are naturally in bijection with the set of circular permutations in Sn, so that one obtains an order on this set
We may assume that w = 1u, and we show that ji is a circular factor of w, which will suffice, by definition of the order on circular permutations and by 6
Summary
In [5], the second-named author defines for each affine Weyl group an affine variety whose integral points are in bijection with the group. In the case of the affine symmetric group Sn, it turns out that the irreducible components are naturally in bijection with the set of circular permutations in Sn (that is, n-cycles), so that one obtains an order on this set. It is this poset, with its three instances, that we study in the present article.
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