On the Hopf algebra of noncommutative symmetric functions in superspace
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866910707484721152 |
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| author | Arcis, Diego González, Camilo Márquez, Sebastián |
| author_facet | Arcis, Diego González, Camilo Márquez, Sebastián |
| contents | We study in detail the Hopf algebra of noncommutative symmetric functions in superspace sNSym, introduced by Fishel, Lapointe and Pinto. We introduce a family of primitive elements of sNSym and extend the noncommutative elementary and power sum functions to superspace. Then, we give formulas relating these families of functions. Also, we introduce noncommutative Ribbon Schur functions in superspace and provide a explicit formula for their product. We show that the dual basis of these function is given by a family of the so--called fundamental quasisymmetric functions in superspace. This allows us to obtain a explicit formula for the coproduct of fundamental quasisymmetric functions in superspace. Additionally, by projecting the noncommutative Ribbon Schur functions in superspace, we define a new basis for the algebra of symmetric functions in superspace. On the other hand, we also show that sNSym can be realised as a Hopf algebra of trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_11813 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Hopf algebra of noncommutative symmetric functions in superspace Arcis, Diego González, Camilo Márquez, Sebastián Combinatorics Rings and Algebras 05E05, 16T05, 16T30 We study in detail the Hopf algebra of noncommutative symmetric functions in superspace sNSym, introduced by Fishel, Lapointe and Pinto. We introduce a family of primitive elements of sNSym and extend the noncommutative elementary and power sum functions to superspace. Then, we give formulas relating these families of functions. Also, we introduce noncommutative Ribbon Schur functions in superspace and provide a explicit formula for their product. We show that the dual basis of these function is given by a family of the so--called fundamental quasisymmetric functions in superspace. This allows us to obtain a explicit formula for the coproduct of fundamental quasisymmetric functions in superspace. Additionally, by projecting the noncommutative Ribbon Schur functions in superspace, we define a new basis for the algebra of symmetric functions in superspace. On the other hand, we also show that sNSym can be realised as a Hopf algebra of trees. |
| title | On the Hopf algebra of noncommutative symmetric functions in superspace |
| topic | Combinatorics Rings and Algebras 05E05, 16T05, 16T30 |
| url | https://arxiv.org/abs/2205.11813 |