• DOI:10.1007/s00031-011-9161-6
• Corpus ID: 973082

Stability inequalities and universal Schubert calculus of rank 2

@article{Berenstein2010StabilityIA,
  title={Stability inequalities and universal Schubert calculus of rank 2},
  author={Arkady Berenstein and Michael Kapovich},
  journal={Transformation Groups},
  year={2010},
  volume={16},
  pages={955-1007}
}

• A. Berenstein, M. Kapovich
• Published 10 August 2010
• Mathematics
• Transformation Groups

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 At 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 At leads to the homology theory H*(Y).