Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras

Authors:
Samuel K. Hsiao;Gizem Karaali
Affiliations:
Mathematics Program, Bard College, Annandale-on-Hudson, USA 12504;Department of Mathematics, Pomona College, Claremont, USA 91711
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2011

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Abstract

We develop a theory of multigraded (i.e., 驴 l -graded) combinatorial Hopf algebras modeled on the theory of graded combinatorial Hopf algebras developed by Aguiar et al. (Compos. Math. 142:1---30, 2006). In particular we introduce the notion of canonical k-odd and k-even subalgebras associated with any multigraded combinatorial Hopf algebra, extending simultaneously the work of Aguiar et al. and Ehrenborg. Among our results are specific categorical results for higher level quasisymmetric functions, several basis change formulas, and a generalization of the descents-to-peaks map.