Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 17 September 2013

Appendix A

A5. Schur-Weyl duality and Young symmetrizers

Let V be the usual n-dimensional representation of GL(n,) on column vectors of length n, that is V=span{b1,,bn} wherebi= (0,,0,1,0,,0)t, and the 1 in bi appears in the ith entry. Then Vk=span { bi1bik |1i1,, ikn } is the span of the words of length k in the letters bi (except that the letters are separated by tensor symbols). The general linear group GL(n,) and the symmetric group Sk acton Vk by g(v1vk) =gv1gvk, and(v1vk) σ=vσ(1) vσ(k), where gGL(n,), σSk, and v1,,vkV. (We have chosen to make the Sk-action a right action here, one could equally well choose the action of Sk to be a left action but then the formula would be σ(v1vk)= vσ-1(1)vσ-1(k).) The following theorem is the amazing relationship between the group Sk and the group GL(n,) which was discovered by Schur [Sch1901] and exploited with such success by Weyl [Wey1946].

Theorem A5.1. (Schur-Weyl duality)

(a) The action of Sk on Vk generates EndGL(n,)(Vk).
(b) The action of GL(n,) on Vk generates EndSk(Vk).

This theorem has the following important corollary, which provides a intimate correspondence between the representation theory of Sk and some of the representations of GL(n,) (the ones indexed by partitions of k).

Corollary A5.2. As GL(n,)×Sk bimodules Vk λkVλ Sλ, where Vλ is the irreducible GL(n,)-module and Sλ is the irreducible Sk-module indexed by λ.

If λ is a partition of k, then the irreducible GL(n,)-representation Vλλ is given by VλVk P(T)N(T), where T is a tableau of shape λ and P(T) and N(T) are as defined in Section 2, Question C.

Notes and references

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..

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