Robinson-Schensted algorithm and Vogan equivalence

We provide a combinatorial proof for the coincidence of Knuth equivalence classes, Kazhdan–Lusztig left cells and Vogan classes for the symmetric group, involving only Robinson-Schensted algorithm and the combinatorial part of the Kazhdan–Lusztig cell theory. The determination of Kazhdan–Lusztig cells for the symmetric group is given in the proof of [4, Thm1.4]. The argument is largely combinatorial except the use of a Vogan’s result [6, Thm6.5] which is obtained in the context of the theory of primitive ideals for universal enveloping algebras. If one can replace this by a combinatorial argument, then it becomes possible to discuss the Kazhdan–Lusztig cell theory for symmetric groups within the scope of these standard texts such as [2] and [5]. This note was motivated by this.