Infinitesimal Hopf algebras and the cd-index of polytopes
type: Item
instance of: scholarly article
title: Infinitesimal Hopf algebras and the cd-index of polytopes
zbMATH Open document ID: 1011.05061
author: O. Diekmann
published in: Discrete \& Computational Geometry
publication date: 2002-05-21T00:00:00Z
review text: A fundamental result in the combinatorial study of polytopes is that their flag \(f\)-vectors satisfy the generalized Dehn-Sommerville relations of \textit{M. M. Bayer} and \textit{L. J. Billera} [Invent. Math. 79, 143-157 (1985; Zbl 0543.52007)]. This is true more generally for the flag \(f\)-vectors of Eulerian posets -- the face lattice of a polytope is an Eulerian poset. The flag \(f\)-vector of a ranked poset is encoded algebraically by its \(\mathbf{ab}\)-index in the algebra \(k\langle{\mathbf a},{\mathbf b}\rangle\) of non-commutative polynomials in \({\mathbf a}\) and \({\mathbf b}\). In this setting, the generalized Dehn-Sommerville relations are equivalent to the flag \(f\)-vector being a member of the subalgebra \(k\langle{\mathbf c},{\mathbf d}\rangle\), where \({\mathbf c}={\mathbf a}+{\mathbf b}\) and \({\mathbf d}={\mathbf ab}+{\mathbf ba}\). The \(\mathbf{cd}\)-index of a polytope is its flag \(f\)-vector, expressed as an element in \(k\langle{\mathbf c},{\mathbf d}\rangle\). Ehrenborg deepened this algebraic study by showing that the association of a flag \(f\)-vector to a graded poset is a morphism of Hopf algebras from a Hopf algebra of posets to the Hopf algebra of quasi-symmetric functions [\textit{R. Ehrenborg}, Adv. Math. 119, 1-25 (1996; Zbl 0851.16033)]. Such Hopf morphisms underlying quasi-symmetric functions associated to combinatorial objects are ubiquitous in enumerative algebraic combinatorics and a general theory was developed to explain this by \textit{N. Bergeron} et al. [J. Comb. Theory, Ser. A 91, 84-110 (2000; Zbl 0969.05064)]. Under a different product, the Hopf algebra of quasi-symmetric functions is isomorphic to \(k\langle{\mathbf a},{\mathbf b}\rangle\) and its coproduct becomes a derivation, a structure that Aguiar studied, called an infinitesimal bialgebra. In this paper, Aguiar puts all these results on a solid algebraic framework, both explaining this ubiquity of quasi-symmetric functions in combinatorics and giving a ne…
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 1743174
zbMATH Keywords: Hopf algebra
zbMATH Keywords: polytope
zbMATH Keywords: Eulerian poset
zbMATH Keywords: Dehn-Sommerville relations
reviewed by: Frank Sottile
MaRDI profile type: Publication
full work available at URL: s00454-001-0049-4
OpenAlex ID: W2073813410
DOI: 10.1007/S00454-001-0049-4
description: scientific article; zbMATH DE number 1743174