  We construct spherical harmonics for fuzzy spheres of even and odd
dimensions, generalizing the correspondence between finite matrix algebras and
fuzzy two-spheres.
  The finite matrix algebras associated with the various fuzzy spheres have a
natural basis which falls in correspondence with tensor constructions of
irreducible representations of orthogonal groups SO(n). This basis is useful in
describing fluctuations around various D-brane constructions of fuzzy spherical
objects. The higher fuzzy spheres are non-associative algebras that appear as
projections of associative algebras related to Matrices. The non-associativity
(as well as the non-commutativity) disappears in the leading large $N$ limit,
ensuring the correct classical limit. Some simple aspects of the combinatorics
of the fuzzy four-sphere can be accounted by a heuristic picture of giant
fractional instantons.
