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

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.