The commutation graph for the longest signed permutation

Using the standard Coxeter presentation for the signed symmetric group $ (B)( )(n+1) on n + 1 letters, two reduced expressions for a given signed permutation are in the same commutation class if one expression can be obtained from the other one by applying a finite sequence of commutations. The commutative classes of a given signed permutation can be seen as the vertices of a graph, called the commutation graph, where two classes are connected by an edge if there are elements in those classes that differ by a long braid relation. We define a rank function for the commutation graph for the longest signed permutation, and use this function to compute the diameter and the radius of the graph. We also prove that the commutation graph for the longest signed permutation is not planar for n > 2, and identify the classes with a single element. (c) 2022 Elsevier B.V. All rights reserved.