Robinson-Schensted correspondence and left cells

This is based on [A]. In [A], I explained several theorems which focused on a famous theorem of [KL] that two elements of the symmetric group belong to a same left cell if and only if they share a common Q-symbol. The first half of [A] was about the direct proof of this theorem (Theorem A), and the second half was about the relation between primitive ideals and left cells, and I explained another proof of this theorem. The reason why I gave a direct proof which was different from the proof in [KL] was that the proof in [KL] was hard to read: It relied on [V1, 6], which in turn relied on [Jo1], the full paper of which is not yet available even today. Note also that the theorem itself is not stated in [KL]. But the beginning part of the proof of [KL, Theorem 1.4] gives some explanation on the relation between left cells in the sense of Kazhdan and Lusztig and Vogan’s generalized τ -invariants in the theory of primitive ideals. In this picture, Theorem A is derived from Joseph’s theorem. Lack of a clear proof in the literature lead Garsia and McLarnan to the publication of [GM]. 1 The proof given in [GM] is close to [A], but the line of the proof in [GM] is interrupted with combinatorics of tableaux, which is not necessary. In fact, after we read to the fourth section of [KL], which is the section for some preliminaries to the proof of [KL, Theorem 1.4], we can give a short and elementary proof of Theorem A in a direct way, as I will show below. I rush to say that my proof was not so original: It copied argument in [Ja1, Satz 5.25] for Joseph’s theorem. This was the reason why I did not publish it in English. But after a decade has passed, we still have no suitable literature which includes the direct proof. Further, we have new development in the last decade. For example, we have better understanding of this theorem in jeu de taquin context (see [H],[BSS]); the study of sovable lattice models in Kyoto school lead to the theory of crystal bases and canonical bases, by which we can understand Theorem A in the crystal base theory context. I have therefore decided to add this short note to this volume in order to give this proof and new development. I also prepare enough papers in the references