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On the Chow Ring of a Flag
Author(s) -
Wenzel Christian
Publication year - 1997
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19971870114
Subject(s) - mathematics , flag (linear algebra) , reductive group , algebraic group , combinatorics , algebraically closed field , borel subgroup , projection (relational algebra) , ring (chemistry) , primitive permutation group , codimension , group (periodic table) , algebraic number , pure mathematics , algebra over a field , mathematical analysis , symmetric group , cyclic permutation , chemistry , organic chemistry , algorithm
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.