On the Chow Ring of a Flag

Abstract

Let G be a reductive linear algebraic group over an algebraically closed field K, let P be a parabolic subgroup scheme of G containing a Borel subgroup B, and let P = Pred ⊂ P be its reduced part. Then P is reduced, a variety, one of the well known classical parabolic subgroups. For char(K) = p > 3, a classification of the P's has been given in [W1]. The Chow ring of G/P only depends on the root system of G. Corresponding to the natural projection from G/P to G/P there is a map of Chow rings from A(G/P) to A(G/P). This map will be explicitly described here. Let P = B, and let p > 3. A formula for the multiplication of elements in A(G/P) will be derived. We will prove that A(G/P) ≃ A(G/P) (abstractly as rings) if and only if G/P ≃ G/P as varieties, i. e., the Chow ring is sensitive to the thickening. Furthermore, in certain cases A(G/P) is not any more generated by the elements corresponding to codimension one Schubert cells.