Diameter of a commutation class on reduced words (Preprint)

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Type: Preprint
National /International: International
Title: Diameter of a commutation class on reduced words
Publication Date: 2023-06-01
Authors: - Gonçalo Gutierres
- Ricardo Mamede
Abstract:

Any permutation w of the symmetric group can be generated by a product of adjacent transpositions, and a reduced word for w is a sequence of generators of minimal length whose product is w. The main result in this paper gives a formula to compute the diameter of a commutation class of the graph G(w), whose vertices are reduced words for w and whose edges are braid relations. To do so, we define a metric on the set of all reduced words of a given permutation which turn out to be equal to the usual distance in any commutation class. If a permutation is fully commutative, i.e. it has only one commutation class, then the formula gives the diameter of G(w). The diameter for a Grassmanian permutation is also given in terms of its Lehman code.

Institution: DMUC 23-17
Online version: http://www.mat.uc.pt...prints/eng_2023.html
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