On the Chow Ring of a Flag

✍ Scribed by Christian Wenzel

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 834 KB
• Volume: 187
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19971870114

✦ Synopsis

Abstract

Let G be a reductive linear algebraic group over an algebraically closed field K, let P̃ be a parabolic subgroup scheme of G containing a Borel subgroup B, and let P = P̃~red~ ⊂ P̃ 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 P̃'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/P̃ there is a map of Chow rings from A(G/P̃) 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/P̃) will be derived. We will prove that A(G/P̃) ≃ A(G/P) (abstractly as rings) if and only if G/P ≃ G/P̃ as varieties, i. e., the Chow ring is sensitive to the thickening. Furthermore, in certain cases A(G/P̃) is not any more generated by the elements corresponding to codimension one Schubert cells.