2017/11/22 by Pawlowski, Brendan, Rhoades, Brendon · 3 citations
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1711.08301
The Delta Conjecture of Haglund, Remmel, and Wilson predicts the monomial expansion of the symmetric function Δ'_ek-1 en, where k ≤ n are positive integers and Δ'_ek-1 is a Macdonald eigenoperator. When k = n, the specialization Δ'_en-1 en|t = 0 is the Frobenius image of the graded Sn-module afforded by the cohomology ring of the \em flag variety consisting of complete flags in ℂn. We define and study a variety Xn,k which carries an action of Sn whose cohomology ring H\bullet(Xn,k) has Frobenius image given by Δ'_ek-1 en|t = 0, up to a minor twist. The variety Xn,k has a cellular decomposition with cells Cw indexed by length n words w = w1 … wn in the alphabet \1, 2, …, k\ in which each letter appears at least once. When k = n, the variety Xn,k is homotopy equivalent to the flag variety. We give a presentation for the cohomology ring H\bullet(Xn,k) as a quotient of the polynomial ring ℤ[x1, …, xn] and describe polynomial representatives for the classes [ Cw] of the closures of the cells Cw; these representatives generalize the classical Schubert polynomials.