Composing arbitrarily many SU(N) fundamentals

Abstract

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.