  The cohomology of the classifying space BU(n) of the unitary groups can be
identified with the ring of symmetric polynomials on n variables by restricting
to the cohomology of BT, where T is a maximal torus in U(n). In this paper we
explore the situation where BT = (CP^{infinity})^n is replaced by a product of
finite dimensional projective spaces (CP^d)^n, fitting into an associated
bundle U(n) x_T (S^{2d+1})^n -> (CP^d)^n -> BU(n). We establish a purely
algebraic version of this problem by exhibiting an explicit system of
generators for the ideal of truncated symmetric polynomials. We use this
algebraic result to give a precise descriptions of the kernel of the
homomorphism in cohomology induced by the natural map (CP^d)^n -> BU(n). We
also calculate the cohomology of the homotopy fiber of the natural map ES_n
x_{S_n} (CP^d)^n -> BU(n).
