Discrete CATS Seminar

Master Exam

Speaker:  Maxwell Hosler, University of Kentucky

Title:  An order on circular permutations

Abstract:

Discussing a paper by Abram, Chapelier-Laget, and Reutenauer, we examine a family a lattices with three isomorphic expressions; first, as a lattice of circular permutations, second, as a lattice of natural-valued functions called 'admitted vectors,' and third, as an interval in the weak order on the affine symmetric group. This family turns out to have strong analogies with the weak order on the symmetric group, despite not being a weak order. Amongst other things, admitted vectors act as 'inversion sets with multiplicity' for  these permutations, and the Hasse diagram can be labelled by transpositions in a way reminiscent of how the same can be done for the weak order. We end by proving the fact that, in some sense, the 'limit' of this family of posets is Young's lattice.