The Graded Dual of a Combinatorial Hopf Algebra on Partition Diagrams

Lingxiao Hao; Shenglin Zhu

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arXiv · 2504.20622

The Graded Dual of a Combinatorial Hopf Algebra on Partition Diagrams

Lingxiao Hao·Shenglin Zhu

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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}.

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BibTeX RIS

Lingxiao Hao, Shenglin Zhu. 2025-11-13. The Graded Dual of a Combinatorial Hopf Algebra on Partition Diagrams. https://arxiv.org/abs/2504.20622

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