The Witt ring of a complex flag variety describes the interestingi.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 G 2 and F 4 . The results also extend to flag varieties over other algebraically closed fields.The Witt ring of a complex flag variety can be approached in two ways. Algebraic geometers might define a complex flag variety as a projective homogeneous variety under some complex reductive group G C . As such, it will have the form G C /P for some parabolic subgroup P . There is a Z/4-graded multiplicative cohomology theory W * alg (−) on algebraic varieties, due to Balmer [Bal05], which extends the usual notion of the Witt ring of quadratic forms over a field. The Witt rings under investigation in this paper are precisely the rings W * alg (G C /P ). From a topological point of view, we may equivalently define a complex flag variety as a quotient manifold G/L obtained from a compact Lie group G by dividing out a Levi subgroup L, i.e. the centralizer of some torus. Indeed, as a manifold, G C /P is diffeomorphic to such a quotient of the maximal compact subgroup G ⊂ G C . Moreover, the Witt ring of G/L can be defined purely topologically as the Z/4-graded ring given in degree i byHere, KO * denotes real topological K-theory, K denotes complex topological K-theory, and the Witt group W i is the quotient of the former by the latter 1
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