Cohomology of $BP{U}_n$ and rings of invariants of Weyl groups

Let $P{U}_n$ denote the projective unitary group of rank $n$ and $BP{U}_n$ be its classifying space, for $n>1$. Using the Serre spectral sequence associated to the fibration $B{U}_n\to BP{U}_n\to K(\mathbb{Z},3)$, we compute the integral cohomology group of $BP{U}_n$ in dimensions $\le 14$. In addition, we determine the ring structure of $H^{\ast }(BP{U}_n;\mathbb{Z})$ up to dimension $13$ by computing the ring of invariants $H^{\ast }(B{T}_{P{U}_n}{)}^W$ of the Weyl group action in dimensions $\le 12$.