Part II - Representation theory

Summary

In this part we describe some uses of tableaux in studying representations of the symmetric group S_n and the general linear groupGL_m(ℂ). We will see that to each partition λ of n one can construct an irreducible representation S^λ of the symmetric group S_n (called a Specht module) and an irreducible representation E^λ of GL(E) for E a finite dimensional complex vector space (called a Schur or Weyl module). The space S^λ will have a basis with one element ν_T for each standard tableau T of shape λ. If e₁, …, e_m is a basis for E, then E^λ will have a basis with one element e_T for each (semistandard) tableau T on λ. with entries from [m]. These basis vectors e_T will be eigenvectors for the diagonal matrix with entries x₁, … x_m, with eigenvalue x^T; the character of the representation will be the Schur polynomial s_λ(x₁, …, x_m).

Two extreme cases of these constructions should be familiar in some version, corresponding to the two extreme partitions λ = (n) and, λ = (1ⁿ). We describe these here in order to fix some notation, as well as to motivate the general story.