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Chow ring of generic flag varieties
Author(s) -
Karpenko Nikita A.
Publication year - 2017
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.201600529
Subject(s) - mathematics , flag (linear algebra) , isomorphism (crystallography) , conjecture , pure mathematics , filtration (mathematics) , ring (chemistry) , combinatorics , epimorphism , simple (philosophy) , algebraic group , variety (cybernetics) , algebraic number , algebra over a field , mathematical analysis , philosophy , chemistry , statistics , organic chemistry , epistemology , crystal structure , crystallography
Let G be a split semisimple algebraic group over a field k and let X be the flag variety (i.e., the variety of Borel subgroups) of G twisted by a generic G ‐torsor. We start a systematic study of the conjecture, raised in [5][N. A. Karpenko, 2017] in form of a question, that the canonical epimorphism of the Chow ring of X onto the associated graded ring of the topological filtration on the Grothendieck ring of X is an isomorphism. Since the topological filtration in this case is known to coincide with the computable gamma filtration, this conjecture indicates a way to compute the Chow ring. We reduce its proof to the case of k = Q . For simply‐connected or adjoint G , we reduce the proof to the case of simple G . Finally, we provide a list of types of simple groups for which the conjecture holds. Besides of some classical types considered previously (namely, A, C, and the special orthogonal groups of types B and D), the list contains the exceptional types G 2 , F 4 , and simply‐connected E 6 .