×

An inductive method of expounding the representation theory of symmetric groups. (English. Russian original) Zbl 0895.20011

Russ. Math. Surv. 51, No. 2, 355-356 (1996); translation from Usp. Mat. Nauk 51, No. 2, 153-154 (1996).
The existing methods of constructing the representation theory of symmetric groups, originating in the work of Young, Schur, Frobenius, and others, use Young diagrams and their combinatorics. The non-fortuitous character of the appearance of the diagrams in this theory is elucidated only at the very end of the construction, when it becomes clear that the branching of the representations \({\mathfrak S}_n|{\mathfrak S}_{n-1}\) coincides with the branching of the Young diagrams with \(n\) cells with respect to diagrams with \(n-1\) cells. The deficiencies of the traditional method (already passed from book to book for nearly 100 years) are evident: firstly, it would be desirable to justify the appearance of the diagrams a priori (instead of a posteriori); secondly, the essential technique does not use the Coxeter property of the groups and cannot be carried over to general Coxeter-Weyl groups; thirdly, the construction lacks an inductive character – it does not use the fact that the \({\mathfrak S}_n\) form a series. It was shown elsewhere that from the Coxeter relations and the distributivity of branching we can deduce the rule of branching itself and find the representation matrices (the Young orthogonal form). In this paper we shall finally get rid of all the remaining restrictions: from only the Coxeter relations, using the so-called Murphy-Jucys (MJ) generators, we shall find the spectrum of the Gel’fand-Tsetlin algebra of the group \({\mathfrak S}_n\), thus realizing the plan of constructing the representation theory of \({\mathfrak S}_n\) outlined elsewhere, and opening the way to carrying the method over to all Coxeter groups and local algebras.

MSC:

20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
PDFBibTeX XMLCite
Full Text: DOI