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UBC Theses and Dissertations

Essential dimension of representations of algebras Scavia, Federico

Abstract

Let k be a field, A be a finitely generated associative k-algebra, and Rep_A[n] be the functor sending a field K containing k to the set of isomorphism classes of representations of A_K of dimension at most n. In the first part of this thesis, 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 goes to infinity. 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. In the second part of the thesis, we study the essential dimension of representations of a fixed quiver with given dimension vector. We also consider the question of when the genericity property holds, i.e., when essential dimension and generic essential dimension agree. We classify the quivers satisfying the genericity property for every dimension vector and show that for every wild quiver the genericity property holds for infinitely many of its Schur roots. We also construct a large class of examples, where the genericity property fails. Our results are particularly detailed in the case of Kronecker quivers.

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Attribution-NonCommercial-NoDerivatives 4.0 International

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