2019/06/06 by D'Adderio, Michele
#05E05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1906.02633
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.