Chow ring of generic flag varieties

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 .