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.

arXiv commentAdditional derivations and material, version published in Nucl. Phys. B; 25 pages, no figures

References in corpus (9)

• Physics and Mathematics of Calogero particles
• Phases of one-dimensional SU(N) cold atomic Fermi gases --from molecular Luttinger liquids to topological phases
• Exact Spectrum of SU(n) Spin Chain with Inverse-Square Exchange
• Equivalence of Two Dimensional QCD and the $c=1$ Matrix Model
• Singular solutions, repeated roots and completeness for higher-spin chains
• Spin Multiplicities
• Composition of many spins, random walks and statistics
• Cumulants, lattice paths, and orthogonal polynomials
• Exact eigenspectrum of the symmetric simple exclusion process on the complete, complete bipartite, and related graphs

Cited by in corpus (5)

• Digitizing lattice gauge theories in the magnetic basis: reducing the breaking of the fundamental commutation relations
• Phase transitions in the decomposition of $SU(N)$ representations
• Nonabelian ferromagnets with three-body interactions
• Counting Bethe States in Twisted Spin Chains
• Adjoint ferromagnets