Commutation Classes of the Reduced Words for the Longest Element of ${\mathfrak S}_n$

Abstract

 Using the standard Coxeter presentation for the symmetric group $\mathfrak{S}_{n}$, two reduced expressions for the same group element $\textsf{w}$ are said to be commutationally equivalent if one expression can be obtained from the other one  by applying a finite sequence of commutations. The commutation classes can be seen as the vertices of a graph $\widehat{G}(\textsf{w})$, where two classes are connected by an edge if elements of those classes differ by a long braid relation. We compute the radius and diameter of the graph $\widehat{G}(\textsf{w}_{\bf 0})$,  for the longest element  $\textsf{w}_{\bf 0}$ in the symmetric group $\mathfrak{S}_{n}$, and show that it is not a planar graph for $n\geq 6$. We also describe a family of commutation classes which contains all atoms, that is classes with one single element, and a subfamily of commutation classes whose elements are in bijection with standard Young tableaux of certain moon-polyomino shapes.