A forest formula for the antipode in incidence Hopf algebras

TL;DR

This paper introduces a novel forest-based formula for calculating the antipode in incidence Hopf algebras, applicable to families of lattices and posets, with conditions for cancellation-free computation.

Contribution

It provides a new combinatorial formula for the antipode in incidence Hopf algebras, extending to various lattice and poset families with characterization of cancellation-free cases.

Findings

Formula expressed as an alternating sum over forests

Map from chains of lattices to forests established

Extension of the formula to poset families with characterization

Abstract

We present a new formula for the antipode of incidence Hopf algebras. This formula is expressed as an alternating sum over forests. First, we prove the formula for incidence Hopf algebras of families of lattices by exhibiting a map from chains of a lattice to forests. Then, we extend the definition and present an analogous formula for the antipode of incidence Hopf algebras of families of posets. We characterize those families for which our formula is cancellation-free.