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Universal rings of invariants

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.