The Witt rings of many flag varieties are exterior algebras

TL;DR

This paper demonstrates that the Witt rings of many complex flag varieties, including exceptional types, are exterior algebras, with generator degrees determined by Dynkin diagram combinatorics, advancing understanding of their KO-theory.

Contribution

It establishes that a broad class of flag varieties have Witt rings as exterior algebras, with generator degrees explicitly linked to Dynkin diagrams, extending previous results.

Findings

Witt rings of many flag varieties are exterior algebras

Generator degrees are determined by Dynkin diagram combinatorics

Results include flag varieties of types G2 and F4

Abstract

The Witt ring of a complex flag variety describes the interesting -- i.e. torsion -- part of its topological KO-theory. We show that for a large class of flag varieties, these Witt rings are exterior algebras, and that the degrees of the generators can be determined by Dynkin diagram combinatorics. Besides full flag varieties, projective spaces, and other varieties whose Witt rings were previously known, this class contains many flag varieties of exceptional types. Complete results are obtained for flag varieties of types G2 and F4. The results also extend to flag varieties over other algebraically closed fields