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arXiv · 1808.02755

On lexicographic representatives in braid monoids

Ramón Flores·Juan González-Meneses
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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 $P_{n,1}\geq \frac{1}{8}$ for every $n\geq 1$. We also provide an explicit formula, based on the Fibonacci numbers, for the number of states of the automaton.

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Ramón Flores, Juan González-Meneses. 2018-08-08. On lexicographic representatives in braid monoids. https://arxiv.org/abs/1808.02755

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