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Using the Lieb-Mattis ordering theorem of electronic energy levels, weidentify the Hilbert space of the low energy sector of U($N$) quantumHall/Heisenberg ferromagnets at filling factor $M$ for $L$ Landau/lattice siteswith the carrier space of irreducible representations of U($N$) described byrectangular Young tableaux of $M$ rows and $L$ columns, and associated withGrassmannian phase spaces U($N$)/U($M$)$\times$U($N-M$). We embed this$N$-component fermion mixture in Fock space through a Schwinger-Jordan (bosonand fermion) representation of U($N$)-spin operators. We provide differentrealizations of basis vectors using Young diagrams, Gelfand-Tsetlin patternsand Fock states (for an electron/flux occupation number in thefermionic/bosonic representation). U($N$)-spin operator matrix elements in theGelfand-Tsetlin basis are explicitly given. Coherent state excitations abovethe ground state are computed and labeled by complex $(N-M)\times M$ matrixpoints $Z$ on the Grassmannian phase space. They adopt the form of a U($N$)displaced/rotated highest-weight vector, or a multinomial Bose-Einsteincondensate in the flux occupation number representation. Replacing U($N$)-spinoperators by their expectation values in a Grassmannian coherent state allowsfor a semi-classical treatment of the low energy (long wavelength)U($N$)-spin-wave coherent excitations (skyrmions) of U($N$) quantum Hallferromagnets in terms of Grasmannian nonlinear sigma models.

Hilbert space structure of the low energy sector of U(N) quantum Hall ferromagnets and their classical limit