Essential dimension of representations of algebras

TL;DR

This paper investigates how the essential dimension of algebra representations grows with dimension, revealing that growth rates classify the algebra's representation type as finite, tame, or wild.

Contribution

It establishes a direct link between the asymptotic growth of essential dimension and the representation type of the algebra, providing a new invariant to distinguish algebras.

Findings

Bounded growth indicates finite representation type.

Linear growth corresponds to tame representation type.

Quadratic growth signifies wild representation type.

Abstract

Let $k$ be a field, $A$ a finitely generated associative $k$-algebra and $\operatorname{Rep}_A[n]$ the functor $\operatorname{Fields}_k\to \operatorname{Sets}$, 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 $r_A(n) := \operatorname{ed}_k(\operatorname{Rep}_A[n])$, 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, $r_A(n)$ is a finer invariant of A, which allows us to distinguish among algebras of the same representation type.