Vanishing and nonvanishing criteria in Schubert calculus

If G′ ↪ G is an inclusion of complex reductive connected Lie groups, there is an induced map H*(G/B) → H*(G′/B′) on the cohomology of flag varieties. The image of a Schubert class is a positive sum of Schubert classes on G′/B′. We investigate the problem of determining which Schubert classes appear in this sum with nonzero coeffecient. This problem plays an important role in representation theory and symplectic geometry. We present a combinatorial game, the “root game,” which provides some criteria for determining which of these coefficients appear. An important special case, addressed by the root game, is the problem of determining when a product Schubert classes on G/B is nonzero. Although the root game is not proven to cover all cases, it works very well in practice, giving a complete answer to the latter question for G = SL(7, C).