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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| Hlavní autoři: | , |
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| Médium: | E-kniha |
| Jazyk: | angličtina |
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Providence, Rhode Island
American Mathematical Society
2023
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| Vydání: | 1 |
| Edice: | Memoirs of the American Mathematical Society |
| Témata: | |
| ISBN: | 9781470467081, 1470467089 |
| ISSN: | 0065-9266, 1947-6221 |
| On-line přístup: | Získat plný text |
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Obsah:
- Introduction -- The Hopf monoid of generalized permutahedra -- Permutahedra, associahedra, and inversion -- Submodular functions, graphs, matroids, and posets -- Characters, polynomial invariants, and reciprocity -- Hypergraphs, simplicial complexes, and building sets
- 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