Combinatorial Hopf algebras of simplicial complexes. (English) Zbl 1350.16026
A Hopf algebra \(\mathcal A\) based on simplicial complexes is here defined; its product is given by disjoint union and its coproduct by extraction of vertices. A cancellation-free formula is given for its antipode. A family of characters makes it a combinatorial Hopf algebra, in the sense defined by Aguiar and Bergeron, for any integer \(s\); this gives morphisms from \(\mathcal A\) into the Hopf algebra of quasi-symmetric functions – in fact, here, of symmetric functions. These morphisms encode informations about colorings of simplicial complexes. Specializations of these symmetric functions give a generalization of Stanley’s \((-1)\)-color theorem. A \(q\)-deformation of these characters is also studied.
Reviewer: Loïc Foissy (Calais)
MSC:
16T30 | Connections of Hopf algebras with combinatorics |
05E45 | Combinatorial aspects of simplicial complexes |
05E05 | Symmetric functions and generalizations |
Keywords:
combinatorial Hopf algebras; quasi-symmetric functions; simplicial complexes; colorings; antipodesReferences:
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