A flag variety for the Delta Conjecture

The Delta Conjecture of Haglund, Remmel, and Wilson predicts the monomial expansion of the symmetric function ${\mathrm{\Delta }}_{e_{k-1}}^{\mathrm{\prime }}{e}_n$, where $k\le n$ are positive integers and ${\mathrm{\Delta }}_{e_{k-1}}^{\mathrm{\prime }}$ is a Macdonald eigenoperator. When $k=n$, the specialization ${\mathrm{\Delta }}_{e_{n-1}}^{\mathrm{\prime }}{e}_n{\mathrm{\mid }}_{t=0}$ is the Frobenius image of the graded $S_n$-module afforded by the cohomology ring of the {\em flag variety} consisting of complete flags in ${\mathbb{C}}^n$. We define and study a variety $X_{n,k}$ which carries an action of $S_n$ whose cohomology ring $H^{\bullet }(X_{n,k})$ has Frobenius image given by ${\mathrm{\Delta }}_{e_{k-1}}^{\mathrm{\prime }}{e}_n{\mathrm{\mid }}_{t=0}$, up to a minor twist. The variety $X_{n,k}$ has a cellular decomposition with cells $C_w$ indexed by length $n$ words $w = w_1 \dots w_n$inthealphabet${1, 2, \dots, k}$ in which each letter appears at least once. When $k=n$, the variety $X_{n,k}$ is homotopy equivalent to the flag variety. We give a presentation for the cohomology ring $H^{\bullet }(X_{n,k})$ as a quotient of the polynomial ring $$\mathbb{Z}$[x_1, \dots, x_n]$anddescribepolynomialrepresentativesfortheclasses$[ \overline{C}_w]$oftheclosuresofthecells$C_w$; these representatives generalize the classical Schubert polynomials.