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arXiv · 1906.02633

$e$-positivity of vertical strip LLT polynomials

Michele D'Adderio
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Abstract

In this article we prove the $e$-positivity of $G_{\mathbfν}[X;q+1]$ when $G_{\mathbfν}[X;q]$ is a vertical strip LLT polynomial. This property has been conjectured by Alexandersson and Panova, and by Garsia, Haglund, Qiu and Romero, and it implies several $e$-positivities conjectured by them and also by Bergeron. We make use of a result of Carlsson and Mellit 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.

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Michele D'Adderio. 2019-06-06. $e$-positivity of vertical strip LLT polynomials. https://arxiv.org/abs/1906.02633

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