The Quasisymmetric Grassmannian

TL;DR

The paper introduces the quasisymmetric Grassmannian, a complex of toric varieties within the Grassmannian, characterized by positroid varieties and a new combinatorial object, with implications for cohomology structure.

Contribution

It constructs the quasisymmetric Grassmannian, describes its equations via the quasisymmetric Johnson graph, and relates its cohomology to quasisymmetric polynomials.

Findings

Identifies the quasisymmetric Grassmannian as a union of positroid varieties.

Provides an affine paving and cohomology ring description.

Introduces the quasisymmetric Johnson graph as a key combinatorial tool.

Abstract

We construct a complex of toric varieties we call the quasisymmetric Grassmannian inside the Grassmannian of $r$-planes in $\mathbb{C}^n$. Each irreducible component is a positroid variety and an $S_n$ translate of a toric Richardson variety of ribbon shape. We describe it as the vanishing locus of equations $\Delta_A\Delta_{A'}=0$ in Pl\"ucker coordinates determined by a new noncrossing combinatorial object we call the quasisymmetric Johnson graph. We give an affine paving, and show that its cohomology ring is a quasisymmetric modification of the Borel presentation of the Grassmannian's cohomology, with fundamental quasisymmetric polynomials playing the role of Schur polynomials.