Hopf Monoids and Generalized Permutahedra
Generalized permutahedra are polytopes that arise in combinatorics, algebraic geometry, representation theory, topology, and optimization. They possess a rich combinatorial structure. Out of this structure we build a Hopf monoid in the category of species. Species provide a unifying framework for or...
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| Main Authors: | , |
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| Format: | eBook |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2023
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| Edition: | 1 |
| Series: | Memoirs of the American Mathematical Society |
| Subjects: | |
| ISBN: | 9781470467081, 1470467089 |
| ISSN: | 0065-9266, 1947-6221 |
| Online Access: | Get full text |
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| Summary: | Generalized permutahedra are polytopes that arise in combinatorics, algebraic geometry, representation theory, topology, and
optimization. They possess a rich combinatorial structure. Out of this structure we build a Hopf monoid in the category of species.
Species provide a unifying framework for organizing families of combinatorial objects. Many species carry a Hopf monoid structure and
are related to generalized permutahedra by means of morphisms of Hopf monoids. This includes the species of graphs, matroids, posets,
set partitions, linear graphs, hypergraphs, simplicial complexes, and building sets, among others. We employ this algebraic structure to
define and study polynomial invariants of the various combinatorial structures.
We pay special attention to the antipode of each
Hopf monoid. This map is central to the structure of a Hopf monoid, and it interacts well with its characters and polynomial invariants.
It also carries information on the values of the invariants on negative integers. For our Hopf monoid of generalized permutahedra, we
show that the antipode maps each polytope to the alternating sum of its faces. This fact has numerous combinatorial consequences.
We highlight some main applications:
We hope our work serves as a quick introduction to the theory of Hopf
monoids in species, particularly to the reader interested in combinatorial applications. It may be supplemented with Marcelo Aguiar and
Swapneel Mahajan’s 2010 and 2013 works, which provide longer accounts with a more algebraic focus. |
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| ISBN: | 9781470467081 1470467089 |
| ISSN: | 0065-9266 1947-6221 |
| DOI: | 10.1090/memo/1437 |