4 - Schur functors and the symmetric group

Summary

The equivalence of categories established in 2.2.7 provides a strong link between the representation theory of the Schur algebra and the polynomial representations of GL_n. However, there is a third player in the game, namely the symmetric group, whose role in this chapter is fundamental.

In Theorem 4.1.1 the symmetric group G = Σ_r enters our considerations as follows. If r ≤ n there is a functor from P_K(n,r) to _KGmod induced by sending a polynomial module to the weight space consisting of vectors on which the element d(t) ∈ T of (1.8), acts as left multiplication by t₁…t_r. This weight space may be viewed as a module for G using 1.5.2(ii). In fact this procedure is a manifestation of a quite general principle; section 4.1 contains a summary of it. To give a flavour of this, suppose we have a K-algebra S and an idempotent e ∈ S; we therefore induce a functor Hom_s(Se, –) : _Smod → _eSemod by sending V to eV. In the special case where S = S_r and r ≤ n there exists a particular e inducing an isomorphism eSe ≅ KG. In this incarnation the Horn functor is known as the Schur functor, and it has a myriad uses. For example Green [1980, §6] showed how to construct the representations of G from a knowledge of the representations of S_r(Г).