Let G be an exceptional Lie group with a maximal torus T ⊂ G. We present the integral cohomology ring H∗(G/T ) by a minimal set of Schubert classes on G/T . This completes the program of determining the integral cohomology of all complete flag manifolds G/T in the context of Schubert calculus. The results have been applied in [DZ3] to construct the integral cohomology of a simple Lie group G in...

Let E/k be a cubic field extension and J a simple exceptional Jordan algebra of degree 3 over k. Then £ is a reducing field of J if and only if E is isomorphic to a (maximal) subfield of some isotope of /. If k has characteristic not 2 or 3 and contains the third roots of unity then every simple exceptional Jordan division algebra of degree 3 over k contains a cyclic cubic subfield. Exceptional...

If a black box simple group is known to be isomorphic to a classical group over a eld of known characteristic, a Las Vegas algorithm is used to produce an explicit isomorphism. This is used to upgrade all nearly linear time Monte Carlo permutation group algorithms to Las Vegas algorithms when the input group has no composition factor isomorphic to an exceptional group of Lie type or a 3{dimensi...