Let \(G\) be a semi-simple linear algebraic group over the complex numbers and let \(B\) be a Borel subgroup. The varieties \(X_ w = \overline {BwB/B}\), as \(w\) varies over the Weyl group \(W\), are called the Schubert varieties of the flag variety \(G/B\). The purpose of the present article is to give an elementary algebraic treatment of the cohomology algebra, over the rational numbers \(\mathbb{Q}\), of the Schubert varieties. A precise description of the cohomology algebra is given and an interesting precise connection with the theory of I. N. Bernstein, I. M. Gel’fand and S. I. Gel’fand [Russ. Math. Surv. 28, No. 3, 1-26 (1973); translation from Usp. Mat. Nauk 28, No. 3(171), 3-26 (1973; Zbl 0286.57025)] for the flag variety, is presented. It is also shown that the description of the cohomology algebra \(H^ \bullet (G/B, \mathbb{Q})\), as the coinvariant algebra of \(W\) extends to Schubert varieties.
For the entire collection see [Zbl 0784.00017].