Essential dimension of representations of algebras (Q2227780)

Summary: Let \(k\) be a field, \(A\) be a finitely generated associative \(k\)-algebra and \(\text{Rep}_A[n]\) 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 \(r_A(n) := \text{ed}_k (\text{Rep}_A [n])\), 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.