Totally nonnegative part of the Peterson variety in Lie type A

TL;DR

This paper proves Rietsch's conjecture that the totally nonnegative part of the Peterson variety in Lie type A is homeomorphic to a cube, using toric geometry methods.

Contribution

It provides a proof of Rietsch's conjecture on the cell decomposition of the totally nonnegative Peterson variety in Lie type A.

Findings

The totally nonnegative Peterson variety $Y_{ ext{ge}0}$ is homeomorphic to a cube.

The proof uses toric geometry techniques.

The result confirms the conjectured cell structure in Lie type A.

Abstract

The Peterson variety (which we denote by $Y$) is a subvariety of the flag variety, introduced by Dale Peterson to describe the quantum cohomology rings of all the partial flag varieties. Motivated by the mirror symmetry for partial flag varieties, Rietsch studied the totally nonnegative part $Y_{\ge0}$ and its cell decomposition. Based on the structure of those cells, Rietsch gave the following conjecture in Lie type A; as a cell decomposed space, $Y_{\ge0}$ is homeomorphic to the cube $[0,1]^{\dim_{\mathbb{C}}Y}$. In this paper, we give a proof of Rietsch's conjecture on $Y_{\ge0}$ in Lie type A by using toric geometry which is closely related to the Peterson variety.