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The Graded Dual of a Combinatorial Hopf Algebra on Partition Diagrams

Lingxiao Hao, Shenglin Zhu

TL;DR

The paper constructs ParQSym, the graded dual of the partition-diagram CHA ParSym, and provides a complete combinatorial Hopf-algebraic framework by dualizing ParSym's product and coproduct. It develops multiple bases (R- and L-, deconcatenation, enriched monomial, enriched q-monomial) and analyzes various gradings and filtrations, including an infinitorial structure via an infinitesimal character η_{ParQSym}. The work also identifies subcoalgebras and Hopf subalgebras (e.g., planar diagrams and matchings) and establishes canonical morphisms to QSym and NSym, enabling transfer of duality and basis-theory results. Overall, the paper extends CHA methods to partition-diagram combinatorics, enriching the algebraic toolkit for graph- and diagram-indexed Hopf algebras with explicit bases and dualities.

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}.

The Graded Dual of a Combinatorial Hopf Algebra on Partition Diagrams

TL;DR

The paper constructs ParQSym, the graded dual of the partition-diagram CHA ParSym, and provides a complete combinatorial Hopf-algebraic framework by dualizing ParSym's product and coproduct. It develops multiple bases (R- and L-, deconcatenation, enriched monomial, enriched q-monomial) and analyzes various gradings and filtrations, including an infinitorial structure via an infinitesimal character η_{ParQSym}. The work also identifies subcoalgebras and Hopf subalgebras (e.g., planar diagrams and matchings) and establishes canonical morphisms to QSym and NSym, enabling transfer of duality and basis-theory results. Overall, the paper extends CHA methods to partition-diagram combinatorics, enriching the algebraic toolkit for graph- and diagram-indexed Hopf algebras with explicit bases and dualities.

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}.
Paper Structure (23 sections, 45 theorems, 240 equations)

This paper contains 23 sections, 45 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. ParSym
  3. A dual CHA construction
  4. The coalgebra structure on partition diagrams
  5. The algebra structure on partition diagrams
  6. ParQSym is a bialgebra
  7. ParQSym is a Hopf algebra
  8. ParQSym is a Combinatorial Hopf algebra
  9. Subcoalgebras and Hopf subalgebras of ParQSym
  10. R-basis of ParSym and L-basis of ParQSym
  11. R-basis of ParSym
  12. L-basis of ParQSym
  13. Other gradings
  14. Grading by irreducibility
  15. Grading by $\otimes$-irreducibility and $\bullet$-irreducibility
  16. ...and 8 more sections

Key Result

Lemma 2.1

ref1 Every non-empty partition diagram $\pi$ can be uniquely written in the form where $l\in \mathbb{N}$, and $\pi_{1},\ldots,\pi_{l}$ are all non-empty $\otimes$-irreducible.

Theorems & Definitions (94)

  • Example 1
  • Lemma 2.1
  • Definition 2.1
  • Example 2
  • Example 3
  • Remark 1
  • Definition 2.2
  • Theorem 2.1: ref1
  • Theorem 2.2
  • Definition 3.1: Coproduct on ParQSym
  • ...and 84 more