ON THE CHOW RING OF A FLAG

Let G be a reductive linear algebraic group over an algebraically clos ed field K, let (P) over tilde be a parabolic subgroup scheme of G con taining a Borel subgroup B, and let P = (P) over tilde(red) subset of (P) over tilde 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) over tilde's has been given in [W1]. The C how ring of GIP only depends on the root system of G. Corresponding to the natural projection from GIP to G/(P) over tilde there is a map of Chow rings from A(G/(P) over tilde) to A(GIP). This map will be expli citly described here. Let P = B, and let p > 3. A formula for the mult iplication of elements in A(G/(P) over tilde) will be derived. We will prove that A (G/(P) over tilde) similar or equal to A(G/P) (abstractl y as rings) if and only if G/P similar or equal to G/(P) over tilde as varieties, i.e., the Chow ring is sensitive to the thickening. Furthe rmore, in certain cases A(GIP) is not any more generated by the elemen ts corresponding to codimension one Schubert cells.