  We compute the multiplicity of the irreducible representations in the
decomposition of the tensor product of an arbitrary number $n$ of fundamental
representations of $SU(N)$, and we identify a duality in the representation
content of this decomposition. Our method utilizes the mapping of the
representations of $SU(N)$ to the states of free fermions on the circle, and
can be viewed as a random walk on a multidimensional lattice. We also derive
the large-$n$ limit and the response of the system to an external non-abelian
magnetic field. These results can be used to study the phase properties of
non-abelian ferromagnets and to take various scaling limits.
