8 - Representations of the general linear group

Summary

The main object of this chapter is to construct and study the irreducible polynomial representations of the general linear group GL_mℂ = GL(E), where E is a complex vector space of dimension m. These can be formed by a basic construction in linear algebra that generalizes a well known construction of symmetric and exterior products; they make sense for any module over a commutative ring. These representations are parametrized by Young diagrams λ with at most m rows, and have bases corresponding to Young tableaux on λ with entries from [m]. They can also be constructed from representations of symmetric groups. Like the latter, these have useful realizations both as subspaces and as quotient spaces of naturally occurring representations, with relations given by quadratic equations. The characters of the representations are given in §8.3. To prove that these give all the irreducible representations we use a bit of the Lie group–Lie algebra story, which is sketched in this setting in §8.2. In the last section we describe some variations on the quadratic equations. In particular, we identify the sum of all polynomial representations with a ring constructed by Deruyts a century ago.

A construction in linear algebra

For any commutative ring R and any R-module E, and any partition λ, we will construct an R-module denoted E^λ. (For applications in these notes, the case where R = ℂ, so E is a complex vector space, will suffice.)