In this paper, we prove a combinatorial rule describing the restriction of any irreducible representation of U(n + m) to the subgroup U(n) × U(m). We also derive similar rules for the reductions from SU(n + m) to S(U (n) × U(m)), and from SU(n + m) to SU(n) × SU(m). As applications of these representationtheoretic results, we compute the spectra of the Bochner-Laplacian on powers of the determinant bundle over the complex Grassmannian Gr n (C n+m ). The spectrum of the Dirac operator acting on the spin Grassmannian Gr n (C n+m ) is also partially computed. A further application is given by the determination of the spectrum of the Hodge-Laplacian acting on the space of smooth functions on the unit determinant bundle over Gr n (C n+m ).

We construct by purely representation-theoretic methods fuzzy versions of an arbitrary complex Grassmannian M = Gr n (C n+m ), i.e., a sequence of matrix algebras tending SU (n + m)-equivariantly to the algebra of smooth functions on M . We also show that this approximation can be interpreted in terms of the Berezin-Toeplitz quantization of M . Furthermore, we use branching rules to prove that the quantization of every complex line bundle over M is given by a SU (n + m)-equivariant truncation of the space of its L 2 -sections.

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Approximating the algebra of complex-valued smooth functions on a space-time manifold by a sequence of matrix algebras [Formula: see text], with dN ↗ ∞, is the basic idea of fuzzy manifolds. In this paper, we explicitly construct fuzzy versions of the homogeneous spaces SO(2n + 1)/U(n) and Sp (n)/U(1) × Sp (n - 1) for n ≥ 2. This allows us to extend a result of Zhang giving a construction of fuzzy irreducible compact Hermitian symmetric spaces to a class of flag manifolds.