The K-theory of versal flags and cohomological invariants of degree 3

TL;DR

This paper provides an explicit description of the Grothendieck ring of certain twisted flag varieties and computes cohomological invariants of degree 3, extending previous results in algebraic K-theory and invariants.

Contribution

It extends Hilbert basis techniques to Laurent polynomials over integers and explicitly describes the Grothendieck ring for specific types of groups, also computing degree 3 invariants.

Findings

Explicit presentation of K_0(X) for G of type A or C

Computation of degree 3 cohomological invariants

Extension of previous results in algebraic invariants

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 ${\rm A}$ or ${\rm C}$ (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.