Yagita's counter-examples and beyond

Abstract

A conjecture on a relationship between the Chow and Grothendieck rings for the generically twisted variety of Borel subgroups in a split semisimple group G, stated by the second author, has been disproved by Nobuaki Yagita in characteristic 0 for G=Spin(2n+1) with n =8 and n=9. For n=8, the second author provided an alternative simpler proof of Yagita's result, working in any characteristic, but failed to do so for n=9. This gap is filled here by involving a new ingredient - Pieri type K-theoretic formulas for highest orthogonal grassmannians. Besides, a similar counter-example for n=10 is produced. Note that the conjecture on Spin(2n+ 1) holds for n up to 5; it remains open for n=6, n=7, and every n >= 11.