Hopf structures in the representation theory of direct products (Q2112577)

Combinatorial Hopf algebras give a linear algebraic structure to infinite families of combinatorial objects. This paper constructs representation-theoretic Hopf algebras out of towers of groups. The basic structure is a supercharacter theory of groups. Applying the supercharacter theory to a direct product of groups builds a family of Hopf algebras. The construction using the simplest possible supercharacters, the group functions which are constant on the class of all non-identities, gives a Hopf algebra isomorphic to the non-commutative symmetric functions NSym. The Hopf algebras also give embeddings into the symmetric functions in non-commuting variables NCSym, and the Malvenuto-Reutenauer Hopf algebra MR. In the general case, we can study the structure of the character groups, and construct the antipode.