New Hopf Structures on Binary Trees
The multiplihedra $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of polytopes tha...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2009-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/2740/pdf |
| _version_ | 1827324071985020928 |
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| author | Stefan Forcey Aaron Lauve Frank Sottile |
| author_facet | Stefan Forcey Aaron Lauve Frank Sottile |
| author_sort | Stefan Forcey |
| collection | DOAJ |
| description | The multiplihedra $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of polytopes that certainly are not: the permutahedra $\mathfrak{S}_{\bullet}$ and associahedra $\mathcal{Y}_{\bullet}$. The maps $\mathfrak{S}_{\bullet} \twoheadrightarrow \mathcal{M}_{\bullet} \twoheadrightarrow \mathcal{Y}_{\bullet}$ reveal several new Hopf structures on tree-like objects nestled between the Hopf algebras $\mathfrak{S}Sym$ and $\mathcal{Y}Sym$. We begin their study here, showing that $\mathcal{M}Sym$ is a module over $\mathfrak{S}Sym$ and a Hopf module over $\mathcal{Y}Sym$. An elegant description of the coinvariants for $\mathcal{M}Sym$ over $\mathcal{Y}Sym$ is uncovered via a change of basis-using Möbius inversion in posets built on the $1$-skeleta of $\mathcal{M}_{\bullet}$. Our analysis uses the notion of an $\textit{interval retract}$ that should be of independent interest in poset combinatorics. It also reveals new families of polytopes, and even a new factorization of a known projection from the associahedra to hypercubes. |
| first_indexed | 2024-04-25T02:02:56Z |
| format | Article |
| id | doaj.art-6a53d23481a04bfe84aa792ce47e6f42 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:56Z |
| publishDate | 2009-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-6a53d23481a04bfe84aa792ce47e6f422024-03-07T14:45:40ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502009-01-01DMTCS Proceedings vol. AK,...Proceedings10.46298/dmtcs.27402740New Hopf Structures on Binary TreesStefan Forcey0Aaron Lauve1Frank Sottile2Department of MathematicsDepartment of Mathematics [Austin]Department of Mathematics [Texas A&M University]The multiplihedra $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of polytopes that certainly are not: the permutahedra $\mathfrak{S}_{\bullet}$ and associahedra $\mathcal{Y}_{\bullet}$. The maps $\mathfrak{S}_{\bullet} \twoheadrightarrow \mathcal{M}_{\bullet} \twoheadrightarrow \mathcal{Y}_{\bullet}$ reveal several new Hopf structures on tree-like objects nestled between the Hopf algebras $\mathfrak{S}Sym$ and $\mathcal{Y}Sym$. We begin their study here, showing that $\mathcal{M}Sym$ is a module over $\mathfrak{S}Sym$ and a Hopf module over $\mathcal{Y}Sym$. An elegant description of the coinvariants for $\mathcal{M}Sym$ over $\mathcal{Y}Sym$ is uncovered via a change of basis-using Möbius inversion in posets built on the $1$-skeleta of $\mathcal{M}_{\bullet}$. Our analysis uses the notion of an $\textit{interval retract}$ that should be of independent interest in poset combinatorics. It also reveals new families of polytopes, and even a new factorization of a known projection from the associahedra to hypercubes.https://dmtcs.episciences.org/2740/pdfhopf algebrasbinary treesassociahedronpermutahedronpermutationsmultiplihedron[math.math-co] mathematics [math]/combinatorics [math.co][info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Stefan Forcey Aaron Lauve Frank Sottile New Hopf Structures on Binary Trees Discrete Mathematics & Theoretical Computer Science hopf algebras binary trees associahedron permutahedron permutations multiplihedron [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | New Hopf Structures on Binary Trees |
| title_full | New Hopf Structures on Binary Trees |
| title_fullStr | New Hopf Structures on Binary Trees |
| title_full_unstemmed | New Hopf Structures on Binary Trees |
| title_short | New Hopf Structures on Binary Trees |
| title_sort | new hopf structures on binary trees |
| topic | hopf algebras binary trees associahedron permutahedron permutations multiplihedron [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2740/pdf |
| work_keys_str_mv | AT stefanforcey newhopfstructuresonbinarytrees AT aaronlauve newhopfstructuresonbinarytrees AT franksottile newhopfstructuresonbinarytrees |