Composing arbitrarily many $SU(N)$ fundamentals

We compute the multiplicity of the irreducible representations in the decomposition of the tensor product of an arbitrary number n of fundamental representations of <math altimg="si1.svg"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math>, and we identify a duality in the representation content of this decomposition. Our method utilizes the mapping of the representations of <math altimg="si1.svg"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> 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.