Vanishing and Non-Vanishing Criteria for Branching Schubert Calculus by

Vanishing and Non-Vanishing Criteria for Branching Schubert Calculus by Kevin Purbhoo Doctor of Philosophy in Mathematics University of California at Berkeley Professor Allen Knutson, Chair We investigate several related vanishing problems in Schubert calculus. First we consider the multiplication problem. For any complex reductive 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. More generally, one can look at vanishing of Schubert intersection numbers, which generalise the multiplication problem to looking at products of more than two classes. 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 the criteria given by the root game are not proven to cover all cases, in practice they work very well, giving a complete answer to the question for G = GL(n,C), n ≤ 7. The root game can be used to study the vanishing problem for multiplication on H(G/P ) (where P ⊂ G is a parabolic subgroup) by pulling back the (G/P )Schubert classes to H(G/B). In the case where G/P is a Grassmannian, the Schubert structure constants are Littlewood-Richardson numbers. We show that the root game gives a necessary and sufficient rule for non-vanishing of Schubert calculus on Grassmannians. A Littlewood-Richardson number is non-zero if and only if it is possible to win the corresponding root game. More generally, the rule