In this article we prove the e-positivity of $G_{\mathit{\nu }}[X;q+1]$ when $G_{\mathit{\nu }}[X;q]$ is a vertical strip LLT polynomial. This property has been conjectured in [2] and [7], and it implies several e-positivities conjectured in those references and in [3].

We make use of a result of Carlsson and Mellit [5] that shows that a vertical strip LLT polynomial can be obtained by applying certain compositions of operators of the Dyck path algebra to the constant 1. Our proof gives in fact an algorithm to expand these symmetric functions in the elementary basis, and it shows, as a byproduct, that these compositions of operators are actually multiplication operators.

在本文中，我们证明了Ë的-positivity$G_{\mathit{\nu }}[X;q+\mathrm{1个}]$ 什么时候 $G_{\mathit{\nu }}[X;q]$是垂直带LLT多项式。在[2]和[7]中已经推测出了该性质，并且暗示了在这些参考文献和[3]中推测出的几个电子阳性。

我们利用Carlsson和Mellit [5]的结果表明，可以通过将Dyck路径代数的某些算符组合应用于常数1来获得垂直带LLT多项式。我们的证明实际上给出了扩展这些的算法对称函数在基本基础上是对称的，并且作为副产品表明，这些算子的组合实际上是乘法算子。