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REPRESENTATION VARIETIES OF ALGEBRAS WITH NODES

Part of: Representation theory of rings and algebras Theory of modules and ideals Special varieties

Published online by Cambridge University Press:  15 June 2021

Ryan Kinser Open the ORCID record for Ryan Kinser [Opens in a new window]  and
András C. Lőrincz
Ryan Kinser
Affiliation:
University of Iowa, Department of Mathematics, Iowa City, IA, USA ([email protected])
András C. Lőrincz
Affiliation:
Humboldt–Universität zu Berlin, Institut für Mathematik, Berlin, Germany ([email protected])
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Abstract

We study the behaviour of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for different algebras, which preserves properties like normality or having rational singularities. Furthermore, we describe how the defining equations of such closed subvarieties change under the correspondence.

By working in the ‘relative setting’ (splitting one node at a time), we demonstrate that there are many nonhereditary algebras whose irreducible components of representation varieties are all normal with rational singularities. We also obtain explicit generators of the prime defining ideals of these irreducible components. This class contains all radical square zero algebras, but also many others, as illustrated by examples throughout the paper. We also show that this is true when irreducible components are replaced by orbit closures, for a more restrictive class of algebras. Lastly, we provide applications to decompositions of moduli spaces of semistable representations of certain algebras.

Keywords

representations of algebrasquiversdeterminantal varietiesdeterminantal idealsrational singularitiesnode splittingmoduli spaces

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets 13C40: Linkage, complete intersections and determinantal ideals 14M12: Determinantal varieties
Secondary: 14M05: Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 2215 - 2245
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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