A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials

TL;DR

This paper provides a new combinatorial formula for the Schur expansion of certain LLT polynomials using graph theory, specifically when the associated graph is triangle-free, and relates the polynomial's degree to graph weights.

Contribution

It introduces a graph-based approach to express LLT polynomials' Schur expansion explicitly for triangle-free cases, extending previous partial results.

Findings

Explicit Schur-positive expansion for triangle-free cases

Largest power of q equals total edge weight of the associated graph

New linear relation among LLT polynomials derived from the graph

Abstract

In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial $G_\lambda(x;q)$ in some special cases. We associate a weighted graph $\Pi$ to $\lambda$ and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of $G_\lambda(x;q)$ whenever $\Pi$ is triangle-free. We also prove that the largest power of $q$ in the LLT polynomial is the total edge weight of our graph.