Universal rings of invariants

TL;DR

This paper introduces a universal combinatorial ring that encodes all invariant rings of algebraic structures on vector spaces, with a rich Hopf algebra structure, and explicitly computes invariants for structures with a single endomorphism.

Contribution

It constructs a universal ring of invariants that specializes to all such rings and endows it with a Hopf algebra structure, extending Zelevinsky's PSH-algebra.

Findings

The universal ring $K[X]$ specializes to all rings of invariants $K[U(W)]^{GL(W)}$.

$K[X]$ has a Hopf algebra structure with coproduct, grading, and inner product.

Explicit calculations of invariants for structures with a single endomorphism.

Abstract

Let $K$ be an algebraically closed field of characteristic zero. Algebraic structures of a specific type (e.g. algebras or coalgebras) on a given vector space $W$ over $K$ can be encoded as points in an affine space $U(W)$. This space is equipped with a $\text{GL}(W)$ action, and two points define isomorphic structures if and only if they lie in the same orbit. This leads to study the ring of invariants $K[U(W)]^{\text{GL}(W)}$. We describe this ring by generators and relations. We then construct combinatorially a commutative ring $K[X]$ which specializes to all rings of invariants of the form $K[U(W)]^{\text{GL}(W)}$. We show that the commutative ring $K[X]$ has a richer structure of a Hopf algebra with additional coproduct, grading, and an inner product which makes it into a rational PSH-algebra, generalizing a structure introduced by Zelevinsky. We finish with a detailed study of $K[X]$ in the case of an algebraic structure consisting of a single endomorphism, and show how the rings of invariants $K[U(W)]^{\text{GL}(W)}$ can be calculated explicitly from $K[X]$ in this case.