Closures in varieties of representations and irreducible components

Goodearl, Kenneth; Huisgen-Zimmermann, Birge

doi:10.2140/ant.2018.12.379

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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

DOI: 10.2140/ant.2018.12.379

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 ${Rep}_{d} (Λ)$ of the $Λ$ -modules with fixed dimension vector $d$ . In this situation, the components of ${Rep}_{d} (Λ)$ are always among the closures $\bar{Rep S}$ , 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 $\bar{Rep S}$ by means of module filtrations “governed by $S$ ”; when $Λ$ is truncated, it pins down the $\bar{Rep S}$ completely.

The analysis of the varieties $\bar{Rep S}$ leads to a novel upper semicontinuous module invariant which provides an effective tool towards the detection of components of ${Rep}_{d} (Λ)$ in general. It detects all components when $Λ$ is truncated.

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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

Vol. 12, No. 2, 2018