A Hopf algebra of subword complexes

TL;DR

This paper develops a Hopf algebra framework for subword complexes, providing explicit formulas and connecting to cluster algebra structures, advancing algebraic combinatorics.

Contribution

It introduces a Hopf algebra structure on subword complexes, including an explicit antipode formula and a sub-Hopf algebra on c-clusters.

Findings

Explicit cancellation-free antipode formula using acyclic orientations

Hopf algebra structure extends to both finite and infinite subword complexes

Induces a non-trivial sub-Hopf algebra on c-clusters in cluster theory

Abstract

We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on $c$-clusters in the theory of cluster algebras.