On Graphs of Sets of Reduced Words

TL;DR

This paper investigates the structure of graphs formed by reduced words of permutations, introducing new formulas and bounds for subgraph structures, braid edges, and commutation edges within these graphs.

Contribution

It provides novel formulas for counting edges and bounds for classes in the graphs of reduced words, advancing understanding of their combinatorial structure.

Findings

New formulas for counting braid and commutation edges

Bounds established for the number of classes in reduced words

Analysis of subgraph structures in the permutation graphs

Abstract

Any permutation in the finite symmetric group can be written as a product of simple transpositions $s_i = (i~i+1)$. For a fixed permutation $\sigma \in \mathfrak{S}_n$ the products of minimal length are called reduced decompositions or reduced words, and the collection of all such reduced words is denoted $\mathcal{R}(\sigma)$. Any reduced word of $\sigma$ can be transformed into any other by a sequence of commutation moves or long braid moves. One area of interest in these sets are the congruence classes defined by using only braid or only commutation relations. The set $\mathcal{R}(\sigma)$ can be drawn as a graph, $G(\sigma)$, where the vertices are the reduced words, and the edges denote the presence of a commutation or braid move between the words. This paper presents new work on subgraph structures in $G(\sigma)$, as well as new formulas to count the number of braid edges and commutation edges in $G(\sigma)$. We also include work on bounds for the number of braid and commutation classes in $\mathcal{R}(\sigma)$.