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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A flag variety for the Delta Conjecture
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by Brendan Pawlowski and Brendon Rhoades PDF
Trans. Amer. Math. Soc. 372 (2019), 8195-8248 Request permission

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
  • 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.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 8195-8248
  • MSC (2010): Primary 05E05, 05E10, 14M15
  • DOI: https://doi.org/10.1090/tran/7918
  • MathSciNet review: 4029695