Vanishing and non-vanishing criteria in Schubert calculus

Abstract

For any complex reductive connected 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=SL(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. More generally 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 the homogeneous spaces. The image of a Schubert class under this map is a positive sum of Schubert classes on G'/B'. We investigate the problem of determining which Schubert classes appear with non-zero coefficient. This is the vanishing problem for branching Schubert calculus, which plays an important role in representation theory and symplectic geometry, as shown in [Berenstein-Sjamaar 2000]. The root game generalises to give a vanishing criterion and a non-vanishing criterion for this problem.