Let $G=U(n)$ and $EU(n)=V_{n}(\mathbb{C}^{\infty})$ the infinit Stiefel manifold. So i think that $BU(n)$ is the infinite Grassmannian $G_r(\mathbb{C}^{\infty})$. We have $H_{BU(n)}(pt)= \mathbb{C}[x_{1}, \cdots, x_{n}]$. But from equivariant theory we have $H_{U(n)}(pt) \simeq H_{T}(pt)^{W}$ (where $T$ is a maximal torus in $U(n)$ and $W$ the Weyl group in $U(n)$). But, because $W$ is the permutation group $S_{n}$ it results $H_{T}(pt)\simeq \mathbb{C}[c_{1}, \cdots, c_{n}]$ where the $c_{i}$ are the simmetric polynomial in the $x_{i}$'s. So che cohomology of $BU(n)$ is isomorphic to $\mathbb{C}[x_{1}, \cdots, x_{n}]$ or $\mathbb{C}[c_{1}, \cdots, c_{n}]$?
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