Operators on symmetric polynomials and applications in computing the cohomology of $BP{U}_n$

This paper studies the integral cohomology ring of the classifying space $BP{U}_n$ of the projective unitary group $P{U}_n$. By calculating a Serre spectral sequence, we determine the ring stucture of $H^{\ast }(BP{U}_n;\mathbb{Z})$ in dimensions $\le 11$. For any odd prime $p$, we also determine the $p$-primary subgroups of $H^i(BP{U}_n;\mathbb{Z})$ in the range $i\le 2p+13$ for $i$ odd and $i\le 4p+8$ for $i$ even. The main technique used in the calculation is applying the theory of Young diagrams and Schur polynomials to certain linear operators on symmetric polynomials.