Root games on Grassmannians (Q883145)

The \textit{root game} was introduced by the author in his previous paper \textit{K.~Purbhoo} [Int. Math. Res. Not. 24590, 1--38 (2006; Zbl 1129.14073)]. It is a combinatorial procedure that can be used to determine whether the product of several Schubert cycles in a generalized flag variety \(G/B\) is non-zero. However, it only gives a sufficient condition for non-vanishing of such a product; it is still not proved whether this condition is necessary. In this paper, the author considers the case of Grassmann varieties. The structure constants of the cohomology ring of a Grassmannian in the Schubert basis are called Littlewood-Richardson numbers. Computing these numbers is a classical, but quite involved problem. The root game can be used to find out whether a Littlewood-Richardson number does not vanish. Even more generally, this procedure allows to say whether the product of an arbitrary number of Schubert cycles is non-zero. The main result of this paper states that in this particular case the condition provided by the root game is necessary and sufficient. One of the advantages of this approach is that it is manifestly symmetric: it is immediately clear from the definitions that reordering of the input Schubert cycles does not affect the result of the root game.