# The $K$-Theory of Versal Flags and Cohomological Invariants of Degree 3

### Sanghoon Baek

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea### Rostislav Devyatov

Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON K1N6N5, Canada### Kirill Zainoulline

Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON K1N6N5, Canada

## Abstract

Let $G$ be a split semisimple linear algebraic group over a field and let $X$ be a generic twisted flag variety of $G$. Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the Grothendieck ring $K_0(X)$ in terms of generators and relations in the case $G=G^{sc}/\mu_2>$ is of Dynkin type A or C (here $G^{sc}$ is the simply-connected cover of $G$); we compute various groups of (indecomposable, semi-decomposable) cohomological invariants of degree 3, hence, generalizing and extending previous results in this direction.

## Cite this article

Sanghoon Baek, Rostislav Devyatov, Kirill Zainoulline, The $K$-Theory of Versal Flags and Cohomological Invariants of Degree 3. Doc. Math. 22 (2017), pp. 1117–1148

DOI 10.4171/DM/589