# Noncommutative unicellular LLT polynomials

## TL;DR

This paper extends the relationship between unicellular LLT polynomials and graph chromatic polynomials to a noncommutative setting, broadening understanding of their algebraic structure.

## Contribution

It introduces a noncommutative version of the $(t-1)$-transform relationship for unicellular LLT polynomials within combinatorial Hopf algebras.

## Key findings

- Established a noncommutative analogue of the $(t-1)$-transform property.
- Connected unicellular LLT polynomials to noncommutative symmetric functions.
- Extended combinatorial Hopf algebra frameworks to include these polynomials.

## Abstract

It is known that unicellular LLT polynomials are related to the quasi-symmetric chromatic polynomials of certain graphs by the $(t-1)$-transform of symmetric functions. We investigate the extension of this transformation to various combinatorial Hopf algebras and prove a noncommutative version of this property.