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

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 $n\ge 2$ we completely determine the degree $3$ invariants of a split semisimple group, the quotient of $({\mathrm{SL}}_2{)}^n$ 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.

Lemma 7.1 and Prop. 7.3 corrected