Braided Species, Hopf Monoids, and the Combinatorics of the Braid Arrangement

Abstract

This paper introduces the category of braided finite sets, denoted Set B m , which is equipped with a natural action of the braid group B m [3, 4]. Within this framework, we develop the theory of braided species, a generalization of classical combinatorial species [1, 2] that incorporates braid group representations. We explore the transport of structures along braids, building on the foundational transport principle of species [26], and analyze the resulting isomorphisms and automorphisms. Crucially, leveraging the seminal framework of Aguiar and Mahajan [5, 6, 7] on Hopf monoids in the category of species, we establish that braided species naturally form a Hopf monoid. This construction provides a new geometric realization of the Hopf monoid associated with hyperplane arrangements [8], specifically tailored to the braid arrangement. Our approach offers a unifying categorical and geometric perspective that bridges classical species theory, braid group actions, and the algebraic combinatorics of hyperplane arrangements, opening new directions for the study of symmetries in combinatorial structures.

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BibTeX

@misc{cedric2026braided,
  title = {{Braided Species, Hopf Monoids, and the Combinatorics of the Braid Arrangement}},
  author = {Pemha Binyam Gabriel Cédric},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05837844v1},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Braided Species, Hopf Monoids, and the Combinatorics of the Braid Arrangement
AU  - Pemha Binyam Gabriel Cédric
PY  - 2026
DA  - 2026/09/07
PB  - HAL
UR  - https://hal.science/hal-05837844v1
AB  - This paper introduces the category of braided finite sets, denoted Set B m , which is equipped with a natural action of the braid group B m [3, 4]. Within this framework, we develop the theory of braided species, a generalization of classical combinatorial species [1, 2] that incorporates braid group representations. We explore the transport of structures along braids, building on the foundational transport principle of species [26], and analyze the resulting isomorphisms and automorphisms. Crucially, leveraging the seminal framework of Aguiar and Mahajan [5, 6, 7] on Hopf monoids in the category of species, we establish that braided species naturally form a Hopf monoid. This construction provides a new geometric realization of the Hopf monoid associated with hyperplane arrangements [8], specifically tailored to the braid arrangement. Our approach offers a unifying categorical and geometric perspective that bridges classical species theory, braid group actions, and the algebraic combinatorics of hyperplane arrangements, opening new directions for the study of symmetries in combinatorial structures.
ER  -

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