### Abstract

We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric functions in superspace.

Original language | English (US) |
---|---|

Pages (from-to) | 144-170 |

Number of pages | 27 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 166 |

DOIs | |

State | Published - Aug 1 2019 |

### Fingerprint

### Keywords

- Hopf algebras
- Quasisymmetric functions
- Superspace
- Symmetric functions

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series A*,

*166*, 144-170. https://doi.org/10.1016/j.jcta.2019.02.016

**Hopf algebra structure of symmetric and quasisymmetric functions in superspace.** / Fishel, Susanna; Lapointe, Luc; Pinto, María Elena.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series A*, vol. 166, pp. 144-170. https://doi.org/10.1016/j.jcta.2019.02.016

}

TY - JOUR

T1 - Hopf algebra structure of symmetric and quasisymmetric functions in superspace

AU - Fishel, Susanna

AU - Lapointe, Luc

AU - Pinto, María Elena

PY - 2019/8/1

Y1 - 2019/8/1

N2 - We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric functions in superspace.

AB - We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric functions in superspace.

KW - Hopf algebras

KW - Quasisymmetric functions

KW - Superspace

KW - Symmetric functions

UR - http://www.scopus.com/inward/record.url?scp=85062540646&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062540646&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2019.02.016

DO - 10.1016/j.jcta.2019.02.016

M3 - Article

VL - 166

SP - 144

EP - 170

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

ER -