A flag variety for the Delta Conjecture
Brendan Pawlowski, Brendon Rhoades

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 , where are positive integers and is a Macdonald eigenoperator. When , the specialization is the Frobenius image of the graded -module afforded by the cohomology ring of the {\em flag variety} consisting of complete flags in . We define and study a variety which carries an action of whose cohomology ring has Frobenius image given by , up to a minor twist. The variety has a cellular decomposition with cells indexed by length words in the alphabet in which each letter appears at least once. When , the variety is homotopy…
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