Theta Maps for Combinatorial Hopf Algebras

Stembridge generalized Schur's $Q$ functions to enriched $P$-partitions and peak algebra. It later appears that the $Q$ functions and peak algebra are odd Hopf subalgebra of symmetric functions and quasi-symmetric functions respectively. We will give a strategy to find the odd Hopf subalgebra of any combinatorial Hopf algebra. The maps from symmetric functions to $Q$ functions and from quasi-symmetric functions to peak algebra, called Theta maps, are combinatorial Hopf morphisms and have their own interests. We are going to develop a general theory for theta maps for many families of combinatorial Hopf algebras. In particular, we will describe Theta maps for the Malvenuto-Reutenaur Hopf algebra whose images are potentially generalizations of $Q$ functions and peak algebra. We will also describe a Hopf algebra of permutations, which has a Theta map whose image is exactly the peak algebra.