For the flag space \(F_n(\mathbb{C}^n)\) show that \(H^*(F_n(\mathbb{C}^n); \mathbb{Z}) \approx \mathbb{Z}[x_1, \ldots, x_n]/(\sigma_1, \ldots, \sigma_n)\) where \(\sigma_i\) is the \(i\)th elementary symmetric polynomial.
Solution
Let \(F_n=F_n(\mathbb C^n)\) be the complete flag manifold, and let \[ L_1,\ldots,L_n\to F_n \] be the tautological line bundles: over a flag their fibers are the successive one-dimensional quotients. Put \[ x_i=c_1(L_i)\in H^2(F_n;\mathbb Z). \] Their direct sum is the trivial bundle \(\underline{\mathbb C}^n\), so \[ \prod_{i=1}^n(1+x_i)=1. \] Equating homogeneous parts gives \[ \sigma_1(x_1,\ldots,x_n)=\cdots= \sigma_n(x_1,\ldots,x_n)=0. \] Thus there is a graded-ring homomorphism \[ \Phi:\mathbb Z[x_1,\ldots,x_n]/(\sigma_1,\ldots,\sigma_n) \longrightarrow H^*(F_n;\mathbb Z). \]
We show it is an isomorphism by comparing the standard flag tower. Forgetting the last line gives an iterated projective-bundle description, and repeated Leray–Hirsch shows that \(H^*(F_n;\mathbb Z)\) is free abelian with Poincaré polynomial \[ \prod_{j=1}^n(1+t^2+\cdots+t^{2(j-1)}). \] Moreover the classes \(x_i\) generate at each projective-bundle stage, so \(\Phi\) is surjective.
On the algebraic side, the elementary symmetric polynomials form the standard regular sequence of symmetric invariants, and reduction successively by the monic relations gives a free abelian group with basis, for example, \[ x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n}, \qquad 0\le a_i\le n-i. \] Its Poincaré polynomial is the same product above (and its total rank is \(n!\)). Hence the surjection \(\Phi\) is an isomorphism: \[ \boxed{H^*(F_n(\mathbb C^n);\mathbb Z) \cong \mathbb Z[x_1,\ldots,x_n]/(\sigma_1,\ldots,\sigma_n).} \]