Stability inequalities and universal Schubert calculus of rank 2

TL;DR

This paper develops a universal Schubert calculus framework for rank 2 dihedral reflection groups, linking cohomology theories of spherical buildings to stability inequalities and generalized triangle inequalities.

Contribution

It introduces a new cohomology theory for spherical buildings of type W, showing its independence from specific buildings and connecting it to coinvariant algebras and universal degenerations.

Findings

Cohomology rings depend only on the reflection group W.

Stability cones solve classification of weighted semistable m-tuples.

Universal algebra A underpins the cohomology and homology theories.

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 and verify that, first, it depends only on W (i.e., all such buildings are "homotopy equivalent'') and second, the cohomology ring is the associated graded of the coinvariant algebra of W under certain filtration. We also construct the dual homology "pre-ring'' of Y. The convex "stability'' cones 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 Kapovich, Leeb and Millson equivalently, they are cut out by the generalized triangle inequalities for thick Euclidean buildings with the Tits boundary Y. Quite remarkably, the cohomology ring is obtained from a certain universal algebra A by a kind of "crystal limit'' that has been previously introduced by Belkale-Kumar for the cohomology of flag varieties and Grassmannians. Another degeneration of A leads to the homology theory of Y.