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Pattern Hopf Algebras.

Raul Penaguiao1

  • 1Institute of Mathematics, University of Zurich, Zurich, Switzerland.

Annals of Combinatorics
|July 5, 2022
PubMed
Summary
This summary is machine-generated.

This study introduces combinatorial presheaves and their connection to Hopf algebras, demonstrating that commutative presheaves yield free Hopf algebras. It also proves the freeness of the Hopf algebra for marked permutations using new factorization theorems.

Keywords:
Free algebrasHopf algebrasMarked permutationsPresheavesSpecies

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Area of Science:

  • Combinatorics
  • Algebraic Combinatorics
  • Abstract Algebra

Background:

  • Combinatorial presheaves, introduced by Aguiar and Mahajan, offer a framework for studying substructures.
  • The algebraic framework of species is adapted to analyze these substructures.
  • Hopf algebras are generated by combinatorial objects with substructure, exemplified by permutations and symmetric functions.

Purpose of the Study:

  • To adapt the algebraic framework of species for combinatorial presheaves.
  • To investigate pattern Hopf algebras arising from commutative combinatorial presheaves.
  • To analyze the Hopf algebra structure of marked permutations and their associated factorization theorems.

Main Methods:

  • Adapting the algebraic framework of species.
  • Defining products and coproducts on functions counting patterns.
  • Utilizing factorization theorems and Lyndon words techniques.

Main Results:

  • Any well-behaved family of combinatorial objects with substructure generates a functorial Hopf algebra.
  • Pattern Hopf algebras from commutative combinatorial presheaves (graphs, posets, generalized permutahedra) are proven to be free.
  • The pattern Hopf algebra for marked permutations, using new factorization theorems, is also shown to be free.

Conclusions:

  • The study establishes a general construction of Hopf algebras from combinatorial presheaves.
  • It demonstrates the freeness of Hopf algebras associated with commutative presheaves and marked permutations.
  • This work provides new insights into algebraic structures in combinatorics and their applications.