Last update: 17 September 2013
Let be the usual representation of on column vectors of length that is and the in appears in the entry. Then is the span of the words of length in the letters (except that the letters are separated by tensor symbols). The general linear group and the symmetric group acton by where and (We have chosen to make the a right action here, one could equally well choose the action of to be a left action but then the formula would be The following theorem is the amazing relationship between the group and the group which was discovered by Schur [Sch1901] and exploited with such success by Weyl [Wey1946].
Theorem A5.1. (Schur-Weyl duality)
|(a)||The action of on generates|
|(b)||The action of on generates|
This theorem has the following important corollary, which provides a intimate correspondence between the representation theory of and some of the representations of (the ones indexed by partitions of
Corollary A5.2. As bimodules where is the irreducible and is the irreducible indexed by
If is a partition of then the irreducible λ is given by where is a tableau of shape and and are as defined in Section 2, Question C.
This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.
Key words and phrases. Algebraic combinatorics, representations.
Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..