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Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions. (English. Russian original) Zbl 1239.52006

Russ. Math. Surv. 66, No. 2, 271-367 (2011); translation from Usp. Mat. Nauk 66, No. 2, 67-162 (2011).
The study of polytopes is a central topic in convex geometry. This survey article is devoted to the problem of flag numbers of convex polytopes, and contains an exposition of results that connect the classical theory of convex polytopes to a variety of mathematical disciplines: algebraic geometry, topology, commutative and homological algebra.
The contents of the article include the following topics. 1. Polytopes: definitions and constructions, the Rota-Hopf algebra, and flag vectors. 2. Quasi-symmetric functions and Hopf algebras: the Ehrenborg map, Leibniz-Hopf algebras, and Lie-Hopf algebras. 3. Lyndon words. 4. Topological realization of Hopf algebras. 5. The structure of the face operator algebra \({\mathcal{D}}\). 6. Flag polynomials: rings of flag vectors, and bipyramid and cone operators. 7. The structure of the ring \({\mathcal{D}}^*\). 8. Multiplicative structure of the rings of flag vectors. 9. Hopf modules and comodules. 10. Universal \(G\)-polynomial: the \(g-\) and \(h-\) polynomials of convex polytopes. 11. Homomorphisms of the rings of convex polytopes. 12. The problem of flag vectors. – The appendices include additional topics like the cd-index of a polytope. An important result included is the fact that the coefficients in the expression of the cd-index of a convex polytope are non-negative integers.

MSC:

52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
05E05 Symmetric functions and generalizations
05E45 Combinatorial aspects of simplicial complexes
06A07 Combinatorics of partially ordered sets
16T05 Hopf algebras and their applications
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
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