On lexicographic representatives in braid monoids

Abstract

The language of maximal lexicographic representatives of elements in the positive braid monoid \(A_n\) with n generators is a regular language. We describe with great detail the smallest finite-state automaton accepting such language and study the proportion of elements of length k whose maximal lexicographic representative finishes with the first generator. This proportion tends to some number \(P_{n,1}\), as k tends to infinity, and we show that \(P_{n,1}\ge \frac{3}{16}=0.1875\) for every \(n\ge 1\). We also provide an explicit formula, based on the Fibonacci numbers, for the number of states of the automaton. Finally, we present the pseudocode of an algorithm which computes the adjacency matrix of the finite-state automaton.