Seminormal Representations of Weyl Groups and Iwahori-Hecke Algebras

Abstract

The purpose of this paper is to describe a general procedure for computing analogues of Young's seminormal representations of the symmetric groups. The method is to generalize the Jucys-Murphy elements in the group algebras of the symmetric groups to arbitrary Weyl groups and Iwahori-Hecke algebras. The combinatorics of these elements allows one to compute irreducible representations explicitly and often very easily. In this paper we do these computations for Weyl groups and Iwahori-Hecke algebras of types A n , B n , D n , G 2 . Although these computations are in reach for types F 4 , E 6 , and E 7 , we shall, in view of the length of the current paper, postpone this to another work. IntroductionThis paper is an attempt to simplify, unify, review, and expand the theory of seminormal representations of the symmetric groups. It is my hope that the material in this paper will be clear enough and general enough to dispel some of the mystique of this subject. I would suggest that the reader begin with section 3, the symmetric group case, and go back and pick up the generalities from sections 1 and 2 as they are needed. This will make In such a subject, where the credit for the discoveries has been traditionally complicated and mixed up, it is di cult to claim credit for much of anything. I believe that my general approach to seminormal representations of Weyl groups and Iwahori-Hecke algebras given in sections 1 and 2 of this paper is new, at least nobody explained it to me before. If this was the structure motivating the work of Hoefsmit H], then he certainly didn't give any hint of it, he produced the seminormal representations out of thin air and then proved that they were correct. I believe that my realization that the Jucys-Murphy elements are coming from the very natural central elements in (2.1) and Proposition (2.4) is new. I also think that before this paper nobody has realized that there is a simple concrete connection (Proposition (2.8)) between Jucys-Murphy type elements in Iwahori-Hecke algebras and Jucys-Murphy elements in group algebras of Weyl groups (even for the special cases). I know that the analogues of the Jucys-Murphy elements in Weyl groups of types B and D will be new to some of the experts and known to others. These Jucys-Murphy elements for types B and D are not new, similar elements appear in the paper of Cherednik Ch], but I was not able to recognize them there until they were pointed out to me by M.L. Nazarov. I extend my thanks to him for this. Some people were asking me for JucysMurphy elements in type G 2 as late as June of 1995. In July 1995 I was told that it was unknown how to quantize the elements of Cherednik, i.e. nd analogues of them in the Iwahori-Hecke algebras of types B and D. Of course, this has been done already in 1974, by Hoefsmit.The idea behind my approach to seminormal representations is as old as representation theory, it is the same method that gives us Gelfand-Tsetlin bases, and that is often used to nd the representations of centralizer algebras and ope...

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