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REPRESENTATION VARIETIES OF ALGEBRAS WITH NODES

Part of: Theory of modules and ideals Representation theory of rings and algebras 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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References

Abeasis, S., Del Fra, A. and Kraft, H., The geometry of representations of ${A}_m$ , Math. Ann. 256(3) (1981), 401–418.CrossRefGoogle Scholar
Assem, I., Simson, D. and Skowroński, A., Elements of the Representation Theory of Associative Algebras , Vol. 1, London Mathematical Society Student Texts, 65 (Cambridge University Press, Cambridge, UK, 2006).CrossRefGoogle Scholar
Auslander, M., Reiten, I. and Smalø, S. O., Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, 36 (Cambridge University Press, Cambridge, UK, 1997). Corrected reprint of the 1995 original.Google Scholar
Babson, E., Huisgen-Zimmermann, B. and Thomas, R., Generic representation theory of quivers with relations, J. Algebra 322(6) (2009), 1877–1918.10.1016/j.jalgebra.2009.05.044CrossRefGoogle Scholar
Barot, M. and Schröer, J., Module varieties over canonical algebras, J. Algebra 246(1) (2001), 175–192.CrossRefGoogle Scholar
Bleher, F. M., Chinburg, T. and Huisgen-Zimmermann, B., The geometry of finite dimensional algebras with vanishing radical square, J. Algebra 425 (2015), 146–178.CrossRefGoogle Scholar
Bobiński, G., Normality of maximal orbit closures for Euclidean quivers, Canad. J. Math. 64(6) (2012), 1222–1247.CrossRefGoogle Scholar
Bobiński, G. and Schröer, J., Algebras with irreducible module varieties I, Adv. Math. 343 (2019), 624–639.CrossRefGoogle Scholar
Bobiński, G. and Schröer, J., Algebras with irreducible module varieties II: Two vertex case, J. Algebra 518 (2019), 384–411.CrossRefGoogle Scholar
Bobiński, G., Algebras with irreducible module varieties III: Birkhoff varieties, Int. Math. Res. Not. IMRN. 2021(4) (2021), 2497–2525.CrossRefGoogle Scholar
Bobiński, G. and Zwara, G., Normality of orbit closures for Dynkin quivers of type ${A}_n$ , Manuscripta Math. 105(1) (2001), 103–109.CrossRefGoogle Scholar
Bobiński, G. and Zwara, G., Schubert varieties and representations of Dynkin quivers, Colloq. Math. 94(2) (2002), 285–309.CrossRefGoogle Scholar
Bondal, A. I., Helices, representations of quivers and Koszul algebras, in Helices and Vector Bundles , London Mathematical Society Lecture Note Series, 148, pp. 75–95 (Cambridge University Press, Cambridge, UK, 1990).CrossRefGoogle Scholar
Bongartz, K., A geometric version of the Morita equivalence, J. Algebra 139(1) (1991), 159–171.CrossRefGoogle Scholar
Bongartz, K., Some geometric aspects of representation theory, in Algebras and Modules , pp. 1–27 (American Mathematical Society, Providence, RI, 1998).Google Scholar
Boutot, J.-F., Singularités rationalles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 65–68.CrossRefGoogle Scholar
Brion, M., Representations of quivers, in Geometric Methods in Representation Theory . I, Séminaires et Congrès, 24, pp. 103–144 (Société mathématique de France, Paris, 2012).Google Scholar
Carroll, A. T., Generic modules for gentle algebras, J. Algebra 437 (2015), 177–201.CrossRefGoogle Scholar
Carroll, A. T. and Chindris, C., Moduli spaces of modules of Schur-tame algebras, Algebr. Represent. Theory 18(4) (2015), 961–976.10.1007/s10468-015-9527-xCrossRefGoogle Scholar
Carroll, A. T., Chindris, C., Kinser, R. and Weyman, J., Moduli spaces of representations of special biserial algebras, Int. Math. Res. Not. IMRN 2020(2) (2020), 403–421.CrossRefGoogle Scholar
Chindris, C., On the invariant theory for tame tilted algebras, Algebra Number Theory 7(1) (2013), 193–214.CrossRefGoogle Scholar
Chindris, C. and Kinser, R., Decomposing moduli of representations of finite-dimensional algebras, Math. Ann. 372(1) (2018), 555–580.CrossRefGoogle Scholar
Chindris, C., Kinser, R. and Weyman, J., Module varieties and representation type of finite-dimensional algebras, Int. Math. Res. Not. IMRN 2020(3) (2015), 631–650.CrossRefGoogle Scholar
Crawley-Boevey, W., Decomposition of Marsden-Weinstein reductions for representations of quivers, Compos. Math. 130(2) (2002), 225–239.CrossRefGoogle Scholar
Crawley-Boevey, W. and Schröer, J., Irreducible components of varieties of modules, J. Reine Angew. Math. 553 (2002), 201–220.Google Scholar
De Concini, C. and Strickland, E., On the variety of complexes, Adv. Math. 41 (1981), 57–77.CrossRefGoogle Scholar
de la Peña, J. A., On the dimension of the module-varieties of tame and wild algebras, Comm. Algebra 19(6) (1991), 1795–1807.CrossRefGoogle Scholar
Derksen, H. and Kemper, G., Computational Invariant Theory : Invariant Theory and Algebraic Transformation Groups, I, Enclopaedia of Mathematical Sciences, 130 (Springer-Verlag, Berlin, 2002).CrossRefGoogle Scholar
Derksen, H. and Weyman, J., On the canonical decomposition of quiver representations, Compos. Math. 133(3) (2002), 245–265.CrossRefGoogle Scholar
Derksen, H. and Weyman, J., The combinatorics of quiver representations, Ann. Inst. Fourier (Grenoble) 61(3) (2011), 1061–1131.CrossRefGoogle Scholar
Derksen, H. and Weyman, J., An Introduction to Quiver Representations , Graduate Studies in Mathematics, 184 (American Mathematical Society, Providence, RI, 2017).CrossRefGoogle Scholar
Gabriel, P. and Roiter, A. V., Representations of Finite-Dimensional Algebras (Springer-Verlag, Berlin, 1997). Reprint of the 1992 English translation from the Russian, with a chapter by Keller, B..CrossRefGoogle Scholar
Geiss, C. and Schröer, J., Varieties of modules over tubular algebras, Colloq. Math. 95(2) (2003), 163–183.CrossRefGoogle Scholar
Goodearl, K. R. and Huisgen-Zimmermann, B., Closures in varieties of representations and irreducible components, Algebra Number Theory 12(2) (2018), 379–410.CrossRefGoogle Scholar
Gorodentsev, A. L. and Rudakov, A. N., Exceptional vector bundles on projective spaces, Duke Math. J. 54(1) (1987), 115–130.CrossRefGoogle Scholar
Hoskins, V., Parallels between moduli of quiver representations and vector bundles over curves, SIGMA Symmetry Integrability Geom.Methods Appl. 14(127) (2018), 46.Google Scholar
Huisgen-Zimmermann, B., Classifying representations by way of Grassmannians, Trans. Amer. Math. Soc. 359(6) (2007), 2687–2719.10.1090/S0002-9947-07-03997-9CrossRefGoogle Scholar
Huisgen-Zimmermann, B., Fine and coarse moduli spaces in the representation theory of finite dimensional algebras, in Expository Lectures on Representation Theory, Contemporary Mathematics, 607, pp. 1–34 (American Mathematical Society, Providence, RI, 2014).Google Scholar
Huisgen-Zimmermann, B. and Goodearl, K. R., Irreducible components of module varieties: Projective equations and rationality, in New Trends in Noncommutative Algebra, Contemporary Mathematics, 562, pp. 141–167 (American Mathematical Society, Providence, RI, 2012).Google Scholar
Huisgen-Zimmermann, B. and Shipman, I., Irreducible components of varieties of representations: The acyclic case, Math. Z. 287(3-4) (2017), 1083–1107.CrossRefGoogle Scholar
Kac, V. G., Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56(1) (1980), 57–92.CrossRefGoogle Scholar
Kac, V. G., Infinite root systems, representations of graphs and invariant theory. II, J. Algebra 78(1) (1982), 141–162.CrossRefGoogle Scholar
Kempf, G. R., Images of homogeneous vector bundles and varieties of complexes, Bull. Amer. Math. Soc. 81(5) (1975), 900–901.CrossRefGoogle Scholar
Kempf, G. R., On the collapsing of homogeneous bundles, Invent. Math. 37(3) (1976), 229–239.CrossRefGoogle Scholar
King, A. D., Moduli of representations of finite-dimensional algebras, Q. J. Math. 45(180) (1994), 515–530.CrossRefGoogle Scholar
Kinser, R., K-polynomials of type A quiver orbit closures and lacing diagrams, in Representations of Algebras , Contemporary Mathematics, 705, pp. 99–114 (American Mathematical Society, Providence, RI, 2018).CrossRefGoogle Scholar
Kinser, R. and Rajchgot, J., Type $A$ quiver loci and Schubert varieties, J. Commut. Algebra 7(2) (2015), 265–301.10.1216/JCA-2015-7-2-265CrossRefGoogle Scholar
Kinser, R. and Rajchgot, J., Type $D$ quiver representation varieties, double Grassmannians, and symmetric varieties, Advances in Mathematics , 376, (2021), 107454.CrossRefGoogle Scholar
Kinser, R. and Weyman, J., Irreducible components of some module varieties via the Springer resolution, Oberwolfach Rep. 11(1) (2014), 475–477.Google Scholar
Lakshmibai, V. and Magyar, P., Degeneracy schemes, quiver schemes, and Schubert varieties, Int. Math. Res. Not. IMRN 1998(12) (1998), 627–640.CrossRefGoogle Scholar
Loc, N. Q. and Zwara, G., Modules and quiver representations whose orbit closures are hypersurfaces, Colloq. Math. 134(1) (2014), 57–74.CrossRefGoogle Scholar
Lőrincz, A. C., Singularities of zero sets of semi-invariants for quivers, arXiv:1509.04170, 2015. To appear in J. Commut. Algebra.Google Scholar
Lőrincz, A. C., Minimal free resolutions of ideals of minors associated to pairs of matrices, Proc. Amer. Math. Soc. 149 (2021), 1857–1873.CrossRefGoogle Scholar
Lőrincz, A. C., On the collapsing of homogeneous bundles in arbitrary characteristic, Preprint, 2020, arXiv:2008.08270.Google Scholar
Lőrincz, A. C. and Weyman, J., Free resolutions of orbit closures of Dynkin quivers, Trans. Amer. Math. Soc. 372 (2019), 2715–2734.CrossRefGoogle Scholar
Martínez-Villa, R., Algebras stably equivalent to $l$ -hereditary, in Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Mathematics, 832, pp. 396–431 (Springer, Berlin, 1980).Google Scholar
Mehta, V. B and Trivedi, V., The variety of circular complexes and $F$ -splitting, Invent. Math. 137 (1999), 449–460.10.1007/s002220050333CrossRefGoogle Scholar
Mehta, V. B and Trivedi, V., Variety of complexes and $F$ -splitting, J. Algebra 215 (1999), 352–365.CrossRefGoogle Scholar
Mnëv, N. E., The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, in Topology and Geometry—Rohlin Seminar, Lecture Notes in Mathematics, 1346, pp. 527–543 (Springer, Berlin, 1988).Google Scholar
Mumford, D., Fogarty, J. and Kirwan, F., Geometric Invariant Theory , 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34 (Springer-Verlag, Berlin, 1994).CrossRefGoogle Scholar
Newstead, P. E., Geometric invariant theory, in Moduli Spaces and Vector Bundles , London Mathematical Society Lecture Note Series, 359, pp. 99–127 (Cambridge University Press, Cambridge, UK, 2009).CrossRefGoogle Scholar
Reineke, M., Moduli of representations of quivers, in Trends in Representation Theory of Algebras and Related Topics , EMS Series of Congress Reports, pp. 589–637 (European Mathematical Society, Zürich, 2008).CrossRefGoogle Scholar
Riedtmann, C., Rutscho, M. and Smalø, S. O., Irreducible components of module varieties: An example, J. Algebra 331 (2011), 130–144.CrossRefGoogle Scholar
Riedtmann, C. and Zwara, G., Orbit closures and rank schemes, Comment. Math. Helv. 88(1) (2013), 55–84.CrossRefGoogle Scholar
Schofield, R., Quiver Representations , CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC (Springer, Cham, Switzerland, 2014).Google Scholar
Schofield, A., General representations of quivers, Proc. Lond. Math. Soc. (3) 65(1) (1992), 46–64.10.1112/plms/s3-65.1.46CrossRefGoogle Scholar
Schröer, J., Varieties of pairs of nilpotent matrices annihilating each other, Comment. Math. Helv. 79(2) (2004), 396–426.CrossRefGoogle Scholar
Strickland, E., On the conormal bundle of the determinantal variety, J. Algebra 75 (1982), 523–537.CrossRefGoogle Scholar
Strickland, E., On the variety of projectors, J. Algebra 106 (1987), 135–147.10.1016/0021-8693(87)90025-1CrossRefGoogle Scholar
Sutar, K., Orbit closures of source-sink Dynkin quivers Int. Math. Res. Not. IMRN 2015(11) (2015), 3423–3444.10.1093/imrn/rnu037CrossRefGoogle Scholar
Timashev, D. A., Homogeneous Spaces and Equivariant Embeddings: Invariant Theory and Algebraic Transformation Groups 8, Encyclopaedia of Mathematical Sciences, 138 (Springer, Heidelberg, Germany, 2011).10.1007/978-3-642-18399-7CrossRefGoogle Scholar
Weyman, J., The equations of conjugacy classes of nilpotent matrices, Invent. Math. 98 (1989), 229–245.CrossRefGoogle Scholar
Weyman, J., Cohomology of Vector Bundles and Syzygies , Cambridge Tracts in Mathematics, 149 (Cambridge University Press, Cambridge, UK, 2003).CrossRefGoogle Scholar
Zwara, G., Degenerations for modules over representation-finite algebras, Proc. Amer. Math. Soc. 127(5) (1999), 1313–1322.10.1090/S0002-9939-99-04714-0CrossRefGoogle Scholar
Zwara, G., Singularities of orbit closures in module varieties, in Representations of Algebras and Related Topics , EMS Series of Congress Reports, pp. 661–725 (European Mathematical Society, Zürich, 2011).CrossRefGoogle Scholar