Theta Maps for Combinatorial Hopf Algebras

This thesis introduces a way to generalize of peak algebra. There are several equivalent denitions for the peak algebra. Stembridge describes it via enriched P-partitions to generalize marked shifted tableaux and Schur's Q functions. Nyman shows that it is a the sum of permutations with the same peak set. Aguiar, Bergeron and Sottile show that the peak algebra is the odd Hopf sub-algebra of quasi symmetric functions using their theory of combinatorial Hopf algebras. In all these cases, there is a very natural and well-behaved Hopf algebra morphism from quasi-symmetric functions or non-commutative symmetric functions to their respective peak algebra, which we call the theta map. This thesis focuses on generalizing the peak algebra by constructing generalized theta maps for an arbitrary combinatorial Hopf algebra. The motivating example of this thesis is the Malvenuto-Reutenauer Hopf algebra of permutations. Our main result is a combinatorial description of all of the theta maps of this Hopf algebra whose images are generalizations of the peak algebra. We also give a criterion to check whether a map is a theta map, and we nd theta maps for Hopf sub-algebras of quasi-symmetric functions. We also show the existence of theta maps for any commutative and cocommutative Hopf algebras. From there, we study the diagonally symmetric functions and diagonally quasi-symmetric functions. Lastly, we describe theta maps for a Hopf algebra V on permutations