Vol. 12, No. 2, 2018

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Closures in varieties of representations and irreducible components

Kenneth R. Goodearl and Birge Huisgen-Zimmermann

Vol. 12 (2018), No. 2, 379–410
Abstract

For any truncated path algebra Λ of a quiver, we classify, by way of representation-theoretic invariants, the irreducible components of the parametrizing varieties Repd(Λ) of the Λ-modules with fixed dimension vector d. In this situation, the components of Repd(Λ) are always among the closures RepS¯, where S traces the semisimple sequences with dimension vector d, and hence the key to the classification problem lies in a characterization of these closures.

Our first result concerning closures actually addresses arbitrary basic finite-dimensional algebras over an algebraically closed field. In the general case, it corners the closures RepS¯ by means of module filtrations “governed by S”; when Λ is truncated, it pins down the RepS¯ completely.

The analysis of the varieties RepS¯ leads to a novel upper semicontinuous module invariant which provides an effective tool towards the detection of components of Repd(Λ) in general. It detects all components when Λ is truncated.

Keywords
varieties of representations, irreducible components, generic properties of representations
Mathematical Subject Classification 2010
Primary: 16G10
Secondary: 14M15, 14M99, 16G20
Milestones
Received: 25 February 2017
Revised: 24 September 2017
Accepted: 18 December 2017
Published: 13 May 2018
Authors
Kenneth R. Goodearl
Department of Mathematics
University of California Santa Barbara
Santa Barbara, CA
United States
Birge Huisgen-Zimmermann
Department of Mathematics
University of California Santa Barbara
Santa Barbara, CA
United States