Essential dimension of representations of algebras

Let $k$ be a field, $A$ a finitely generated associative $k$-algebra and ${\mathrm{Rep}}_A[n]$ the functor $\operatorname{Fields}_k\to \operatorname{Sets}$,whichsendsafield$K$containing$k$ to the set of isomorphism classes of representations of $A_K$ of dimension at most $n$. We study the asymptotic behavior of the essential dimension of this functor, i.e., the function $r_A(n):={\mathrm{ed}}_k({\mathrm{Rep}}_A[n])$, as $n\to \mathrm{\infty }$. In particular, we show that the rate of growth of $r_A(n)$ determines the representation type of $A$. That is, $r_A(n)$ is bounded from above if $A$ is of finite representation type, grows linearly if $A$ is of tame representation type and grows quadratically if A is of wild representation type. Moreover, $r_A(n)$ is a finer invariant of A, which allows us to distinguish among algebras of the same representation type.