Stefan Forcey ; Aaron Lauve ; Frank Sottile - New Hopf Structures on Binary Trees

dmtcs:2740 - Discrete Mathematics & Theoretical Computer Science, January 1, 2009, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) -
New Hopf Structures on Binary Trees

Authors: Stefan Forcey 1; Aaron Lauve 2; Frank Sottile 3

  • 1 Department of Mathematics
  • 2 Department of Mathematics [Austin]
  • 3 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.

Volume: DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)
Section: Proceedings
Published on: January 1, 2009
Imported on: January 31, 2017
Keywords: 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]
    Source : OpenAIRE Graph
  • Applicable Algebraic Geometry: Real Solutions, Applications, and Combinatorics; Funder: National Science Foundation; Code: 0701050

Linked publications - datasets - softwares

Source : ScholeXplorer IsRelatedTo ARXIV 2106.16130
Source : ScholeXplorer IsRelatedTo DOI 10.1007/s00026-022-00598-z
Source : ScholeXplorer IsRelatedTo DOI 10.48550/arxiv.2106.16130
  • 2106.16130
  • 10.1007/s00026-022-00598-z
  • 10.48550/arxiv.2106.16130
Diameter estimates for graph associahedra

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