A horizontal-strip LLT polynomial is determined by its weighted graph

TL;DR

This paper establishes that horizontal-strip LLT polynomials are uniquely determined by their associated weighted graphs, leading to new relations and connections with chromatic symmetric functions.

Contribution

It proves that isomorphic weighted graphs imply equal LLT polynomials, defining LLT polynomials indexed by graphs and exploring their relations and connections.

Findings

Horizontal-strip LLT polynomials are determined by their weighted graphs.

New relations between LLT polynomials are established.

Connection with extended chromatic symmetric functions is explored.

Abstract

We prove that two horizontal-strip LLT polynomials are equal if the associated weighted graphs defined by the author in a previous paper are isomorphic. This provides a sufficient condition for equality of horizontal-strip LLT polynomials and yields a well-defined LLT polynomial indexed by a weighted graph. We use this to prove some new relations between LLT polynomials and we explore a connection with extended chromatic symmetric functions.