THE WITT RINGS OF MANY FLAG VARIETIES ARE EXTERIOR ALGEBRAS

. 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 a few well-studied examples such as full flag varieties and projective spaces, this class includes many flag varieties whose Witt rings were previously unknown, including many flag varieties of exceptional types. In particular, it includes all flag varieties of types G2 and F4. The results also extend to flag varieties over other algebraically closed fields.