A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions

TL;DR

This paper provides a new combinatorial characterization and explicit positive expansion of vertical-strip LLT polynomials in elementary symmetric functions, linking graph orientations and chromatic quasisymmetric functions.

Contribution

It introduces a novel combinatorial characterization of LLT polynomials and derives explicit positive formulas in elementary symmetric functions, connecting graph orientations and symmetric functions.

Findings

Explicit combinatorial expansion of LLT polynomials in elementary symmetric functions

Manifest positivity when replacing q with q+1

New bijective proofs relating LLT polynomials and graph colorings

Abstract

We give a new characterization of the vertical-strip LLT polynomials $\mathrm{LLT}_P(x;q)$ as the unique family of symmetric functions that satisfy certain combinatorial relations. This characterization is then used to prove an explicit combinatorial expansion of vertical-strip LLT polynomials in terms of elementary symmetric functions. Such formulas were conjectured independently by A. Garsia et al. and the first named author, and are governed by the combinatorics of orientations of unit-interval graphs. The obtained expansion is manifestly positive if $q$ is replaced by $q+1$, thus recovering a recent result of M. D'Adderio. Our results are based on linear relations among LLT polynomials that arise in the work of D'Adderio, and of E. Carlsson and A. Mellit. To some extent these relations are given new bijective proofs using colorings of unit-interval graphs. As a bonus we obtain a new characterization of chromatic quasisymmetric functions of unit-interval graphs.