The Graded Dual of a Combinatorial Hopf Algebra on Partition Diagrams
Rings and Algebras
2025-11-14 v4
Abstract
John M. Campbell constructed a combinatorial Hopf algebra (CHA) \text{ParSym} on partition diagrams by lifting the CHA structure of \text{NSym} (the Hopf algebra of noncommutative symmetric functions) through an analogous approach. In this article, we define \text{ParQSym}, which is the graded dual of \text{ParSym}. Its CHA structure is defined in an explicit, combinatorial way, by analogy with that of the CHA \text{QSym} of quasisymmetric functions. And we give some subcoalgebra and Hopf subalgebras of \text{ParQSym}, some gradings and filtrations of \text{ParSym} and \text{ParQSym}, and some bases of \text{ParSym} and \text{ParQSym} by analogy with some distinguished bases of \text{NSym} and \text{QSym}.
Cite
@article{arxiv.2504.20622,
title = {The Graded Dual of a Combinatorial Hopf Algebra on Partition Diagrams},
author = {Lingxiao Hao and Shenglin Zhu},
journal= {arXiv preprint arXiv:2504.20622},
year = {2025}
}