Essential dimension of representations of algebras

Abstract

Let $k$ be a field, $A$ be a finitely generated associative $k$-algebra and Rep${}_A[n]$ be the functor Fields${}_k\to$ Sets, which sends a field $K$ 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 $r_A(n):={\mathrm{ed}}_k$ (Rep${}_A[n])$, as $n\to \infty$. 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, $r_A(n)$ allows us to construct invariants of algebras which are finer than the representation type.