The Quasisymmetric Grassmannian

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 ${\mathrm{\Delta }}_A{\mathrm{\Delta }}_{A^{\mathrm{\prime }}}=0$ in Plücker 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.