Invariants that are Covering Spaces and their Hopf Algebras

Abstract

In a previous paper by the author a universal ring of invariants for algebraic structures of a given type was constructed. This ring is a polynomial algebra that is generated by certain trace diagrams. It was shown that this ring admits the structure of a rational positive self adjoint Hopf algebra (abbreviated rational PSH-algebra), and was conjectured that it always admits a lattice that is a PSH-algebra, a structure that was introduced by Zelevinsky. In this paper we solve this conjecture, showing that the universal ring of invariants splits as the tensor product of rational PSH-algebras that are either polynomial algebras in a single variable, or admit a lattice that is a PSH-algebra. We do so by considering diagrams as topological spaces, and using tools from the theory of covering spaces. As an application we derive a formula that connects Kronecker coefficients with finite index subgroups of free groups and representations of their Weyl groups, and a formula for the number of conjugacy classes of finite index subgroup in a finitely generated group that admits a surjective homomorphism onto the group of integers.