Commutation classes of the reduced words for the longest element of \(\mathfrak{S}_{n}\) (Q2185211)

Summary: Using the standard Coxeter presentation for the symmetric group \(\mathfrak{S}_{n} \), two reduced expressions for the same group element \(\mathsf{w}\) 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 \(\widehat{G}(\mathsf{w})\), 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 \(\widehat{G}(\mathsf{w}_0)\), for the longest element \(\mathsf{w}_0\) in the symmetric group \(\mathfrak{S}_{n} \), and show that it is not a planar graph for \(n\geqslant 6\). 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.