# Essential dimension of representations of algebras

### Federico Scavia

The University of British Columbia, Vancouver, Canada

## Abstract

Let $k$ be a field, $A$ be a finitely generated associative $k$-algebra and Rep$_{A}[n]$ be the functor Fields$_{k}→$ 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 $r_{A}(n):=ed_{k}$ (Rep$_{A}[n])$, as $n→∞$. 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.

## Cite this article

Federico Scavia, Essential dimension of representations of algebras. Comment. Math. Helv. 95 (2020), no. 4, pp. 661–702

DOI 10.4171/CMH/500