Invariants that are covering spaces and their Hopf algebras

TL;DR

This paper proves that the universal ring of invariants for algebraic structures can be decomposed into simpler components using topological methods, and connects these invariants to group theory and representation theory.

Contribution

It confirms that the universal ring of invariants splits into tensor products of rational PSH-algebras, advancing understanding of their algebraic and topological structure.

Findings

Universal ring splits into tensor products of rational PSH-algebras.

Provides formulas linking Kronecker coefficients with subgroup structures.

Derives counts of conjugacy classes in finitely generated groups.

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 answer 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.