Combinatorial Hopf algebras from restriction species with preorder cuts
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908965258919936 |
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| author | Fløystad, Gunnar |
| author_facet | Fløystad, Gunnar |
| contents | We get new Hopf algebras (HA): 1. A wealth of quotient HA's of the Malvenuto-Reutenauer HA (the Loday-Ronco HA being a special case). They consist of the permutations avoiding an {\it arbitrary} set of permutations without global descents, 2. A HA of pairs of parking filtrations, and 3. Four HA of pairs of preorders.
New concepts in this setting are: 1. a category Set$_{\mathbb N}$ whose objects are sets, but morphisms are represented by matrices of natural numbers, and 2. restriction species ${\mathsf S}$ on sets coming with pairs of natural transformations $π_1, π_2 : {\mathsf S} \rightarrow$ Pre to the species of preorders. These induce two coproducts $Δ_1$ and $Δ_2$. Dualizing $Δ_1$ gives product $μ_1$ and coproduct $Δ_2$, giving bimonoid species. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_06479 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Combinatorial Hopf algebras from restriction species with preorder cuts Fløystad, Gunnar Rings and Algebras Combinatorics Primary: 16T30, Secondary: 05E99, 06A11 We get new Hopf algebras (HA): 1. A wealth of quotient HA's of the Malvenuto-Reutenauer HA (the Loday-Ronco HA being a special case). They consist of the permutations avoiding an {\it arbitrary} set of permutations without global descents, 2. A HA of pairs of parking filtrations, and 3. Four HA of pairs of preorders. New concepts in this setting are: 1. a category Set$_{\mathbb N}$ whose objects are sets, but morphisms are represented by matrices of natural numbers, and 2. restriction species ${\mathsf S}$ on sets coming with pairs of natural transformations $π_1, π_2 : {\mathsf S} \rightarrow$ Pre to the species of preorders. These induce two coproducts $Δ_1$ and $Δ_2$. Dualizing $Δ_1$ gives product $μ_1$ and coproduct $Δ_2$, giving bimonoid species. |
| title | Combinatorial Hopf algebras from restriction species with preorder cuts |
| topic | Rings and Algebras Combinatorics Primary: 16T30, Secondary: 05E99, 06A11 |
| url | https://arxiv.org/abs/2301.06479 |