A flag variety for the Delta Conjecture

Abstract

The Delta Conjecture of Haglund, Remmel, and Wilson predicts the monomial expansion of the symmetric function Δ e k − 1 ′ e n \Delta ’_{e_{k-1}} e_n , where k ≤ n k \leq n are positive integers and Δ e k − 1 ′ \Delta ’_{e_{k-1}} is a Macdonald eigenoperator. When k = n k = n , the specialization Δ e n − 1 ′ e n | t = 0 \Delta ’_{e_{n-1}} e_n|_{t = 0} is the Frobenius image of the graded S n S_n -module afforded by the cohomology ring of the flag variety consisting of complete flags in C n \mathbb {C}^n . We define and study a variety X n , k X_{n,k} which carries an action of S n S_n whose cohomology ring H ∙ ( X n , k ) H^{\bullet }(X_{n,k}) has Frobenius image given by Δ e k − 1 ′ e n | t = 0 \Delta ’_{e_{k-1}} e_n|_{t = 0} , up to a minor twist. The variety X n , k X_{n,k} has a cellular decomposition with cells C w C_w indexed by length n n words w = w 1 … w n w = w_1 \dots w_n in the alphabet { 1 , 2 , … , k } \{1, 2, \dots , k\} in which each letter appears at least once. When k = n k = n , the variety X n , k X_{n,k} is homotopy equivalent to the flag variety. We give a presentation for the cohomology ring H ∙ ( X n , k ) H^{\bullet }(X_{n,k}) as a quotient of the polynomial ring Z [ x 1 , … , x n ] \mathbb {Z}[x_1, \dots , x_n] and describe polynomial representatives for the classes [ C ¯ w ] [ \overline {C}_w] of the closures of the cells C w C_w ; these representatives generalize the classical Schubert polynomials.