New invariants for permutations, orders and graphs. (English) Zbl 1462.16036
One of the results in the theory of combinatorial Hopf algebras is a a canonical way of constructing combinatorial invariants of Hopf algebras on isomorphism classes of graphs with values in the space QSym of quasisymmetric functions [M. Aguiar et al., Compos. Math. 142, No. 1, 1–30 (2006; Zbl 1092.05070)].
One direction of generalization is to replace the Hopf algebra of graphs with a Hopf algebra of a more general class of objects containing graphs (Hopf algebra structures on permutations and posets). This construction allows to prove “that the chromatic symmetric function and many other invariants have a property we call positively h-alternating”. Also they “show that certain properties, including the Schur and e-positivity, are shared by many of the symmetric function invariants coming from the CHAs defined in this paper. ”
One direction of generalization is to replace the Hopf algebra of graphs with a Hopf algebra of a more general class of objects containing graphs (Hopf algebra structures on permutations and posets). This construction allows to prove “that the chromatic symmetric function and many other invariants have a property we call positively h-alternating”. Also they “show that certain properties, including the Schur and e-positivity, are shared by many of the symmetric function invariants coming from the CHAs defined in this paper. ”
Reviewer: Dmitry Artamonov (Moskva)
MSC:
| 16T30 | Connections of Hopf algebras with combinatorics |
| 05E05 | Symmetric functions and generalizations |
Citations:
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