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On generically split generic flag varieties
Author(s) -
Karpenko Nikita A.
Publication year - 2018
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12161
Subject(s) - mathematics , flag (linear algebra) , algebraic group , borel subgroup , filtration (mathematics) , variety (cybernetics) , combinatorics , quotient , linear algebraic group , group (periodic table) , reductive group , pure mathematics , algebraic number , algebra over a field , group theory , mathematical analysis , chemistry , statistics , organic chemistry
Abstract Let G be a split semisimple algebraic group over an arbitrary field F , let E be a G ‐torsor over F , and let P be a parabolic subgroup of G . The quotient variety X : = E / P , known as a flag variety , is generically split , if the parabolic subgroup P is special. It is generic , provided that the G ‐torsor E over F is a standard generic G k ‐torsor for a subfield k ⊂ F and a split semisimple algebraic group G k over k with( G k ) F = G . For any generically split generic flag variety X , we show that the Chow ringCH X is generated by Chern classes (of vector bundles over X ). This implies that the topological filtration on the Grothendieck ring of X coincides with the computable gamma filtration. The results were already known in some cases including the case where P is a Borel subgroup. We also provide a complete classification of generically split generic flag varieties and, equivalently, of special parabolic subgroups for split simple groups.