Stability inequalities and universal Schubert calculus of rank 2

Abstract

The goal of the paper is to introduce a version of Schubert calculus for each dihedral reflection group W. That is, to each “sufficiently rich” spherical building Y of type W we associate a certain cohomology theory $H_{BK}^*(Y)$ and verify that, first, it depends only on W (i.e., all such buildings are “homotopy equivalent”), and second, $H_{BK}^*(Y)$ is the associated graded of the coinvariant algebra of W under certain filtration. We also construct the dual homology “pre-ring” on Y. The convex “stability” cones in ${\left( {{\mathbb{R}^2}} \right)^m}$ defined via these (co)homology theories of Y are then shown to solve the problem of classifying weighted semistable m-tuples on Y in the sense of [KLM1]; equivalently, they are cut out by the generalized triangle inequalities for thick Euclidean buildings with the Tits boundary Y. The independence of the (co)homology theory of Y refines the result of [KLM2], which asserted that the Stability Cone depends on W rather than on Y. Quite remarkably, the cohomology ring $H_{BK}^*(Y)$ is obtained from a certain universal algebra A _t by a kind of “crystal limit” that has been previously introduced by Belkale–Kumar for the cohomology of ag varieties and Grassmannians. Another degeneration of A _t leads to the homology theory H _*(Y).