  The following thesis contains results on the combinatorial representation
theory of the finite Hecke algebra $H_n(q)$.
  In Chapter 2 simple combinatorial descriptions are given which determine when
a Specht module corresponding to a partition $\lambda$ is irreducible. This is
done by extending the results of James and Mathas. These descriptions depend on
the crystal of the basic representation of the affine Lie algebra
$\widehat{\mathfrak{sl}_\ell}$. In Chapter 3 these results are extended to
determine which irreducible modules have a realization as a Specht module. To
do this, a new condition of irreducibility due to Fayers is combined with a new
description of the crystal from Chapter 2. In Chapter 4 a bijection of cores
first described by myself and Monica Vazirani is studied in more depth. Various
descriptions of it are given, relating to the quotient
$\widetilde{S_\ell}/{S_\ell}$ and to the bijection given by Lapointe and Morse.
