Cohomology classes of interval positroid varieties and a conjecture of Liu

TL;DR

This paper investigates the cohomology classes of diagram varieties associated with subsets of a2^2, disproves Liu's conjecture relating these classes to Specht modules, and provides new formulas for their computation using Stanley symmetric functions.

Contribution

The paper provides a counterexample to Liu's conjecture and introduces a new Schur-positive formula for cohomology classes of interval positroid varieties.

Findings

Counterexample to Liu's conjecture

Upper bound relation between c4 and c3 cohomology classes

New formula for cohomology classes as Stanley symmetric functions

Abstract

To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture. However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians.