Hopf Monoids and Generalized Permutahedra / Marcelo Aguiar and Federico Ardila.

Format:

Book

Author/Creator:

Aguiar, Marcelo, 1962- author.

Ardila, Federico, author.

Series:

Memoirs of the American Mathematical Society ; Volume 289.

Memoirs of the American Mathematical Society Series ; Volume 289

Language:

English

Subjects (All):

Combinatorial analysis.

Physical Description:

1 online resource (132 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

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.

Contents:

Cover

Title page

Introduction

Hopf monoids and generalized permutahedra

Application A. Antipode formulas

Application B. Character theory and reciprocity theorems

Application C. Inversion of formal power series

Outline

Future directions

Acknowledgments

Chapter 1. The Hopf monoid of generalized permutahedra

1.1. A brief guide to Hopf monoids in species

1.2. \rG,\rM,\rP,\rPi,\rF: Graphs, matroids, posets, set partitions, partitions into paths

1.3. Generalized permutahedra

1.4. \rGP: The Hopf monoid of generalized permutahedra

1.5. Maximality of \rGP

1.6. The antipode of \rGP

Chapter 2. Permutahedra, associahedra, and inversion

2.1. The group of characters of a Hopf monoid

2.2. \rbPi: Permutahedra and the multiplication of power series

2.3. \Kcb(\rbA): Associahedra and the composition of power series

2.4. Inversion of formal power series and Loday's question

Chapter 3. Submodular functions, graphs, matroids, and posets

3.1. \rSF: Submodular functions and generalized permutahedra

3.2. \rG: Graphs and graphic zonotopes

3.3. \rM: Matroids and matroid polytopes

3.4. \rP: Posets and poset cones

Chapter 4. Characters, polynomial invariants, and reciprocity

4.1. Invariants of Hopf monoids and reciprocity

4.2. The basic character and the basic invariant of \wGP

4.3. Combinatorial reciprocity for graphs, matroids, and posets

Chapter 5. Hypergraphs, simplicial complexes, and building sets

5.1. \rHGP: Minkowski sums of simplices, hypergraphs, Rota's question

5.2. \rHG: Hypergraphs

5.3. \rSC: Simplicial complexes, graphs, and Benedetti et al.'s formula

5.4. \rBS: Building sets and nestohedra

5.5. \rW: Simple graphs, ripping and sewing, and graph associahedra

5.6. \rPi: Set partitions and permutahedra, revisited.

5.7. \rF: Paths and associahedra, revisited

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

Other Format:

Print version: Aguiar, Marcelo Hopf Monoids and Generalized Permutahedra

ISBN:

1-4704-7592-8