ABSTRACT
We will now explore one of the central motivating results for this book. The set of classical results of representation theory discussed in this chapter will be the inspiration for much of the remaining chapters. At the heart of these results sits the notion of covariant space of a group of (orthogonal) n × n matrices G. One possible explanation for the role of this space stems from the study of the G-module decomposition of the space R[x] of polynomials in n variables. The story begins with the observation that we can trivially get many copies of any known irreducible component V of R[x], simply by tensoring V with a given G-invariant polynomial. This naturally raises the question of finding a basic set of irreducible components out of which all others can be obtained through this means. This is precisely what the G-covariant space will give us in one stroke. For more on this fascinating subject see [Kane 01], or [Garsia and Haiman 08].
