Exploring a Delta Schur Conjecture

IF 0.4 Q4 MATHEMATICS, APPLIED
A. Garsia, J. Liese, J. Remmel, Meesue Yoo
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引用次数: 1

Abstract

In \cite{HRW15}, 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 \cite{GHRY} 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 \cite{GHRY}. 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 \cite{GHRY} we conjectured such a construction mechanism. We prove here that in the case that $\nu=(m-k,1^k)$ with $1\le m< n$ the equality of the two sides can be established by the same methods used in \cite{GHRY}. While this work was in progress, we learned that Rhodes 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 \cite{GHRY}. 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.
探索Delta舒尔猜想
在\cite{HRW15}中,Haglund, Remmel, Wilson提出了一个猜想,该猜想预测了获得对称函数的纯组合方法$\Delta_{e_k}e_n$。它被称为Delta猜想。最近在\cite{GHRY}中证明了Delta猜想在$q=0$或$t=0$中任一情况下成立。在本文中,我们完成了由Remmel发起的一项工作,其最初目的是通过\cite{GHRY}中开发的相同方法探索对称函数$\Delta_{s_\nu} e_n$。这里我们首先需要的是构造对称函数的方法,该函数可以被看作是$t=0$的对称函数$\Delta_{s_\nu} e_n$的“组合侧”。基于\cite{GHRY}的发现,我们推测了这样一种构造机制。我们在这里证明了在$\nu=(m-k,1^k)$的情况下,对于$1\le m< n$,可以用与\cite{GHRY}相同的方法来建立两边的等式。当这项工作进行时,我们了解到Rhodes和Shimozono之前也构建了这样的“组合面”。最近,Jim Haglund从\cite{GHRY}的结果中证明了他们的猜想。本文通过对Haglund论证的适当修改,证明了多项式$\Delta_{s_\nu}e_n$和Rhoades-Shimozono“组合边”具有hook Schur函数展开的多能性评价。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Combinatorics
Journal of Combinatorics MATHEMATICS, APPLIED-
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