A flag variety for the Delta Conjecture

TL;DR

This paper introduces a new variety $X_{n,k}$ that models the Delta Conjecture's symmetric functions through its cohomology, generalizing flag varieties and providing explicit polynomial representatives for its classes.

Contribution

It defines the variety $X_{n,k}$ with an $S_n$-action whose cohomology encodes the Delta Conjecture, extending classical flag variety results and providing a new geometric framework.

Findings

Cohomology ring of $X_{n,k}$ is presented as a quotient of a polynomial ring.

Cell decomposition of $X_{n,k}$ indexed by words with specific properties.

Polynomial representatives generalize Schubert polynomials.

Abstract

The Delta Conjecture of Haglund, Remmel, and Wilson predicts the monomial expansion of the symmetric function $\Delta'_{e_{k-1}} e_n$, where $k \leq n$ are positive integers and $\Delta'_{e_{k-1}}$ is a Macdonald eigenoperator. When $k = n$, the specialization $\Delta'_{e_{n-1}} e_n|_{t = 0}$ 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 $\Delta'_{e_{k-1}} e_n|_{t = 0}$, up to a minor twist. The variety $X_{n,k}$ has a cellular decomposition with cells $C_w$ indexed by length $n$ words $w = w_1 \dots w_n$ in the alphabet $\{1, 2, \dots, k\}$ 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 $\mathbb{Z}[x_1, \dots, x_n]$ and describe polynomial representatives for the classes $[ \overline{C}_w]$ of the closures of the cells $C_w$; these representatives generalize the classical Schubert polynomials.