On $H^{\ast }(BP{U}_n;\mathbb{Z})$ and Weyl group invariants

For the projective unitary group $P{U}_n$ with a maximal torus $T_{P{U}_n}$ and Weyl group $W$, we show that the integral restriction homomorphism $${\rho }_{P{U}_n}\text{\hspace{0.17em}⁣}:H^{\ast }(BP{U}_n;\mathbb{Z})\to H^{\ast }(B{T}_{P{U}_n};\mathbb{Z}{)}^W$$ to the integral invariants of the Weyl group action is onto. We also present several rings naturally isomorphic to $H^{\ast }(B{T}_{P{U}_n};\mathbb{Z}{)}^W$. In addition we give general sufficient conditions for the restriction homomorphism ${\rho }_G$ to be onto for a connected compact Lie group $G$.