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Transactions of the American Mathematical Society

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A flag variety for the Delta Conjecture


Authors: Brendan Pawlowski and Brendon Rhoades
Journal: Trans. Amer. Math. Soc. 372 (2019), 8195-8248
MSC (2010): Primary 05E05, 05E10, 14M15
DOI: https://doi.org/10.1090/tran/7918
Published electronically: September 12, 2019
MathSciNet review: 4029695
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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 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.


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Additional Information

Brendan Pawlowski
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90027
MR Author ID: 1053722
Email: bpawlows@usc.edu

Brendon Rhoades
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093
MR Author ID: 779261
Email: bprhoades@math.ucsd.edu

Keywords: Flag variety, symmetric function, coinvariant algebra
Received by editor(s): January 16, 2018
Received by editor(s) in revised form: June 2, 2019
Published electronically: September 12, 2019
Additional Notes: The second author was partially supported by NSF grant DMS-1500838.
Article copyright: © Copyright 2019 American Mathematical Society