We develop a group-theoretical approach to describe $N$-component composite bosons as planar electrons attached to an odd number $f$ of Chern-Simons flux quanta. This picture arises when writing the Coulomb exchange interaction as a quantum Hall ferromagnet in terms of collective $U(N)$-spin operators. A spontaneously chosen ground state of $M$ electrons per Landau site breaks the symmetry from $U(N)$ to the stability subgroup $U(M)times U(N-M)$, so that coherent state excitations are labeled by points on the Grassmannian coset $U(N)/U(M)times U(N-M)$. The quantization of this Grassmann phase space corresponds to the carrier Hilbert space of unitary irreducible representations of $U(N)$ described by rectangular Young tableaux of $M$ rows and $f$ columns. We construct an embedding of the Hilbert space into Fock space by using a Schwinger realization of collective $U(N)$-spin operators as bilinear products of composite boson operators. We also build a system of Grassmann coherent states and discuss the classical limit of $U(N)$ quantum Hall ferromagnets in terms of nonlinear sigma models on Grasmannians.
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