Twisted gamma filtration and algebras with orthogonal involution

TL;DR

This paper introduces a new torsion element in the gamma-ring of twisted flag varieties, generalizing previous work, and applies it to study algebras with orthogonal involutions, advancing understanding of algebraic structures with involutions.

Contribution

It constructs a non-trivial torsion element in the gamma-ring of twisted flag varieties using PGO-torsors, extending prior results to new algebraic contexts.

Findings

Constructed a non-trivial torsion element in gamma-rings of twisted flag varieties.

Generalized previous constructions from HSpin to broader cases.

Applied torsion elements to analyze algebras with orthogonal involutions.

Abstract

For the Grothendieck group of a split simple linear algebraic group, the twisted gamma-filtration provides a useful tool for constructing torsion elements in gamma-rings of twisted flag varieties. In this paper, we construct a non-trivial torsion element in the gamma-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline. We use this torsion element to study algebras with orthogonal involutions.