A vanishing and a non-vanishing condition for Schubert calculus on G/B

For any complex semisimple Lie group G, many of the structure constants of the ordinary cohomology ring H∗(G/B;Z) vanish in the Schubert basis, and the rest are strictly positive. We present a combinatorial game, the “root game” which provides some criteria for determining which of the Schubert intersection numbers vanish. The definition of the root game is manifestly invariant under automorphisms of G, and under permutations of the classes intersected. Although these criteria are not proven to cover all cases, in practice they work very well, giving a complete answer to the question for G = GL(7,C). In a separate paper we show that one of these criteria is in fact necessary and sufficient when the classes are pulled back from a Grassmannian. 1 Basic Set-up 1.