
arXiv: 1505.04458
We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their $f$-vectors. We also use characters to give a generalization of Stanley’s $(-1)$-color theorem. Nous considérons une algèbre de Hopf de complexes simpliciaux et fournissons une formule sans multiplicité pour son antipode. On obtient ensuite une famille d'algèbres de Hopf combinatoires en définissant une famille de caractères sur cette algèbre de Hopf. Les caractères de ces algèbres de Hopf donnent lieu à des fonctions symétriques qui encode de l’information sur les coloriages du complexe simplicial ainsi que son vecteur-$f$. Nousallons également utiliser des caractères pour donner une généralisation du théorème $(-1)$ de Stanley.
combinatorial Hopf algebras, Symmetric functions and generalizations, Combinatorial Hopf algebra, colorings, Connections of Hopf algebras with combinatorics, combinatorial hopf algebra, quasi-symmetric functions, [info.info-dm] computer science [cs]/discrete mathematics [cs.dm], antipodes, Combinatorial aspects of simplicial complexes, simplicial complexes, [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], QA1-939, FOS: Mathematics, simplicial complex, Mathematics - Combinatorics, Combinatorics (math.CO), 16T30, 05E05, 05E45, Mathematics
combinatorial Hopf algebras, Symmetric functions and generalizations, Combinatorial Hopf algebra, colorings, Connections of Hopf algebras with combinatorics, combinatorial hopf algebra, quasi-symmetric functions, [info.info-dm] computer science [cs]/discrete mathematics [cs.dm], antipodes, Combinatorial aspects of simplicial complexes, simplicial complexes, [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], QA1-939, FOS: Mathematics, simplicial complex, Mathematics - Combinatorics, Combinatorics (math.CO), 16T30, 05E05, 05E45, Mathematics
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