Chow groups of products of Severi-Brauer varieties and invariants of degree 3

TL;DR

This paper investigates degree 3 invariants of certain algebraic groups using Chow groups of Severi-Brauer varieties, providing complete classifications for specific split semisimple and reductive groups.

Contribution

It completely determines degree 3 invariants for quotients of multiple SL(2) groups and extends methods to other types, advancing understanding of invariants via Chow group torsion.

Findings

Classified degree 3 invariants for quotients of ( ext{SL}_2)^n.

Connected Chow group torsion to invariants of algebraic groups.

Provided an example for type A groups using modified methods.

Abstract

We study the semi-decomposable invariants of a split semisimple group and their extension to a split reductive group by using the torsion in the codimension $2$ Chow groups of a product of Severi-Brauer varieties. In particular, for any $n\geq 2$ we completely determine the degree $3$ invariants of a split semisimple group, the quotient of $(\operatorname{\mathbf{SL}}_{2})^{n}$ by its maximal central subgroup, as well as of the corresponding split reductive group. We also provide an example illustrating that a modification of our method can be applied to find the semi-decomposable invariants of a split semisimple group of type A.