Journal of Combinatorics

Volume 10 (2019)

Special Issue in Memory of Jeff Remmel, Part 2 of 2

Guest Editor: Nicholas A. Loehr

Exploring a Delta Schur Conjecture

Pages: 633 – 653

DOI: https://dx.doi.org/10.4310/JOC.2019.v10.n4.a2

Authors

Adriano Garsia (Department of Mathematics, University of California at San Diego)

Jeffrey Liese (Department of Mathematics, California Polytechnic State University, San Luis Obispo, Cal., U.S.A.)

Jeffrey B. Remmel (Department of Mathematics, University of California at San Diego)

Meesue Yoo (Applied Algebra and Optimization Research Center, Sungkyunkwan University, Suwon, South Korea)

Abstract

In [8], Haglund, Remmel, Wilson state a conjecture which predicts a purely combinatorial way of obtaining the symmetric function $\Delta_{e_k} e_n$. It is called the Delta Conjecture. It was recently proved in [1] that the Delta Conjecture is true when either $q = 0$ or $t = 0$. In this paper we complete a work initiated by Remmel whose initial aim was to explore the symmetric function $\Delta_{s_{\nu}} e_n$ by the same methods developed in [1]. Our first need here is a method for constructing a symmetric function that may be viewed as a “combinatorial side” for the symmetric function $\Delta_{s_{\nu}} e_n$ for $t = 0$. Based on what was discovered in [1] we conjectured such a construction mechanism. We prove here that in the case that $\nu= (m - k, 1^k)$ with $1 \leq m \lt n$ the equality of the two sides can be established by the same methods used in [1]. While this work was in progress, we learned that Rhoades and Shimozono had previously constructed also such a “combinatorial side”. Very recently, Jim Haglund was able to prove that their conjecture follows from the results in [1]. We show here that an appropriate modification of the Haglund arguments proves that the polynomial $\Delta_{s_{\nu}} e_n$ as well as the Rhoades–Shimozono “combinatorial side” have a plethystic evaluation with hook Schur function expansion.

The first named author was supported by NSF grant DMS1700233.

The fourth author was supported by NRF grants 2016R1A5A1008055 and 2017R1C1B2005653.