Arkiv för Matematik

Volume 61 (2023)

Number 1

Yagita’s counter-examples and beyond

Pages: 1 – 17



Sanghoon Baek (Department of Mathematical Sciences, KAIST, Daejeon, South Korea)

Nikita A. Karpenko (Mathematical & Statistical Sciences, University of Alberta, Edmonton, AB, Canada)


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 $G=\operatorname{Spin}(2n+1)$ 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 $\operatorname{Spin}(2n+1)$ holds for $n$ up to $5$; it remains open for $n=6$, $n=7$, and every $n \geq 11$.


algebraic groups, generic torsors, projective homogeneous varieties, Chow groups

2010 Mathematics Subject Classification

14C25, 20G15

The work of the first author was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1901-02. This paper has been accomplished during the second author’s stay at the Max-Planck Institute for Mathematics in Bonn.

Received 12 December 2021

Received revised 16 June 2022

Accepted 28 June 2022

Published 26 April 2023