A Hopf algebra generalization of the symmetric functions in partially commutative variables
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929630575853568 |
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| author | Daugherty, Spencer |
| author_facet | Daugherty, Spencer |
| contents | The quasisymmetric functions, $QSym$, are generalized for a finite alphabet $A$ by the colored quasisymmetric functions, $QSym_A$, in partially commutative variables. Their dual, $NSym_A$, generalizes the noncommutative symmetric functions, $NSym$, through a relationship with a Hopf algebra of trees. We define an algebra $Sym_A$, contained within $QSym_A$, that is isomorphic to the symmetric functions, $Sym$, when $A$ is an alphabet of size one. We show that $Sym_A$ is a Hopf algebra and define its graded dual, $PSym_A$, which is the commutative image of $NSym_A$ and also generalizes $Sym$. The seven algebras listed here can be placed in a commutative diagram connected by Hopf morphisms. In addition to defining generalizations of the classic bases of the symmetric functions to $Sym_A$ and $PSym_A$, we describe multiplication, comultiplication, and the antipode in terms of a basis for both algebras. We conclude by defining a pair of dual bases that generalize the Schur functions and listing open questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11013 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Hopf algebra generalization of the symmetric functions in partially commutative variables Daugherty, Spencer Combinatorics 05E05, 16T30 The quasisymmetric functions, $QSym$, are generalized for a finite alphabet $A$ by the colored quasisymmetric functions, $QSym_A$, in partially commutative variables. Their dual, $NSym_A$, generalizes the noncommutative symmetric functions, $NSym$, through a relationship with a Hopf algebra of trees. We define an algebra $Sym_A$, contained within $QSym_A$, that is isomorphic to the symmetric functions, $Sym$, when $A$ is an alphabet of size one. We show that $Sym_A$ is a Hopf algebra and define its graded dual, $PSym_A$, which is the commutative image of $NSym_A$ and also generalizes $Sym$. The seven algebras listed here can be placed in a commutative diagram connected by Hopf morphisms. In addition to defining generalizations of the classic bases of the symmetric functions to $Sym_A$ and $PSym_A$, we describe multiplication, comultiplication, and the antipode in terms of a basis for both algebras. We conclude by defining a pair of dual bases that generalize the Schur functions and listing open questions. |
| title | A Hopf algebra generalization of the symmetric functions in partially commutative variables |
| topic | Combinatorics 05E05, 16T30 |
| url | https://arxiv.org/abs/2412.11013 |