Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras

TL;DR

This paper develops a multigraded combinatorial Hopf algebra framework, introducing canonical subalgebras and extending previous theories to higher levels, with applications to quasisymmetric functions and descent-to-peak maps.

Contribution

It extends the theory of combinatorial Hopf algebras to multigraded settings and introduces canonical k-odd and k-even subalgebras, broadening the scope of prior work.

Findings

Defined canonical k-odd and k-even subalgebras for multigraded Hopf algebras

Derived basis change formulas for these algebras

Generalized the descents-to-peaks map to higher levels

Abstract

We develop a theory of multigraded (i.e., $N^l$-graded) combinatorial Hopf algebras modeled on the theory of graded combinatorial Hopf algebras developed by Aguiar, Bergeron, and Sottile [Compos. Math. 142 (2006), 1--30]. 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.