Finite parabolic conjugation on varieties of nilpotent matrices. (English) Zbl 1327.14258
Let \({\mathcal N}_n^{(x)}\) be the closed variety of \(n\times n\) complex \(x\)-nilpotent matrices; \({\mathcal N}_n^{(x)}\) consists of all matrices \(N\) with the property \(N^x=0\). The paper under review studies the action by conjugation of an arbitrary upper-block parabolic subgroup \(P\) of the general linear group \(\mathrm{GL}_n({\mathbb C})\). This action is translated to a representation-theoretic context. As a result the author obtains a criterion whether the action admits a finite number of orbits. For 2-nilpotent matrices a system of representatives of the finite number of orbits is obtained and a set-theoretic description of their closures is presented. The action of the Borel subgroup is studied in more detail when the minimal degenerations are specified. Further, the author clarifies the finite case of a maximal parabolic action on 3-nilpotent matrices. Finally, she proves that in all non-finite cases, the corresponding quiver algebra is of wild representation type.
Reviewer: Vesselin Drensky (Sofia)
MSC:
| 14R20 | Group actions on affine varieties |
| 16G20 | Representations of quivers and partially ordered sets |
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
| 16W22 | Actions of groups and semigroups; invariant theory (associative rings and algebras) |
References:
| [1] | Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006). doi:10.1017/CBO9780511614309 · Zbl 1092.16001 · doi:10.1017/CBO9780511614309 |
| [2] | Bongartz, K.: Minimal singularities for representations of Dynkin quivers. Comment. Math. Helv. 69(4), 575-611 (1994). doi:10.1007/BF02564505 · Zbl 0832.16008 · doi:10.1007/BF02564505 |
| [3] | Bongartz, K., Gabriel, P.: Covering spaces in representation-theory. Invent. Math. 65(3), 331-378 (1981/82) · Zbl 0482.16026 |
| [4] | Boos, M.: Conjugation on varieties of nilpotent matrices. PhD thesis, Bergische Universität, Wuppertal (2012). http://elpub.bib.uni-wuppertal.de/servlets/DerivateServlet/Derivate-3019/dc1207.pdf · Zbl 1327.14258 |
| [5] | Boos, M., Reineke, M.: B-orbits of 2-nilpotent matrices. In: Highlights in Lie Algebraic Methods (Progress in Mathematics). Birkhäuser, Boston, pp. 147-166 (2011) · Zbl 1248.14054 |
| [6] | Brion, M.: Spherical varieties. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, 1994). Birkhäuser, Boston, pp. 753-760 (1995) · Zbl 0862.14031 |
| [7] | Drozd, Y.A.: Tame and wild matrix problems. In: Representation Theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832. Springer, Berlin, pp. 242-258 (1980) · Zbl 0454.16014 |
| [8] | Fresse, L.: On the singular locus of certain subvarieties of Springer fibres. To appear in: Mathematical Research Letters (2012) · Zbl 1312.14109 |
| [9] | Gabriel, P.: Unzerlegbare Darstellungen. I. Manuscripta Math. 6, 71-103; correction, ibid. 6(1972):309 (1972) · Zbl 0232.08001 |
| [10] | Gabriel, P.: The universal cover of a representation-finite algebra. In: Representations of Algebras (Puebla, 1980), Lecture Notes in Math, vol. 903. Springer, Berlin, pp. 68-105 (1981) · Zbl 0822.14024 |
| [11] | Gerstenhaber, M.: On nilalgebras and linear varieties of nilpotent matrices. III. Ann. Math. 70(2), 167-205 (1959) · Zbl 0168.28103 · doi:10.2307/1969896 |
| [12] | Hesselink, W.: Singularities in the nilpotent scheme of a classical group. Trans. Am. Math. Soc. 222, 1-32 (1976) · Zbl 0332.14017 · doi:10.1090/S0002-9947-1976-0429875-8 |
| [13] | Hille, L., Röhrle, G.: A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical. Transform. Groups 4(1), 35-52 (1999). doi:10.1007/BF01236661 · Zbl 0924.20035 · doi:10.1007/BF01236661 |
| [14] | Jordan, M.E. C.: Sur la résulotion des équations différentielles linéaires. Ouevres 4, 313-317 (1871) |
| [15] | Jordan, M.E.C.: Traité des substitutions et des équations algébriques. Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. Éditions Jacques Gabay, Sceaux (1989). Reprint of the 1870 original · Zbl 0837.14037 |
| [16] | Kraft, H.: Geometrische Methoden in der Invariantentheorie. Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig (1984) · Zbl 0569.14003 |
| [17] | Magyar, P., Weyman, J.M., Zelevinsky, A.: Multiple flag varieties of finite type. Adv. Math. 141, 97-118 (1999) · Zbl 0951.14034 · doi:10.1006/aima.1998.1776 |
| [18] | Melnikov, A.: B-orbits in solutions to the equation X2=0 in triangular matrices. J. Algebra 223(1), 101-108 (2000). doi:10.1006/jabr.1999.8056 · Zbl 0963.15013 · doi:10.1006/jabr.1999.8056 |
| [19] | Melnikov, A.: Description of B-orbit closures of order 2 in upper-triangular matrices. Transform. Groups 11(2), 217-247 (2006). doi:10.1007/s00031-004-1111-0 · Zbl 1196.14040 · doi:10.1007/s00031-004-1111-0 |
| [20] | Panyushev, D.I.: Complexity and nilpotent orbits. Manuscripta Math. 83(3-4), 223-237 (1994). doi:10.1007/BF02567611 · Zbl 0822.14024 · doi:10.1007/BF02567611 |
| [21] | Timashev, D.A.: A generalization of the Bruhat decomposition. Izv. Ross. Akad. Nauk Ser. Mat. 58(5), 110-123 (1994). doi:10.1070/IM1995v045n02ABEH001643 · Zbl 0837.14037 · doi:10.1070/IM1995v045n02ABEH001643 |
| [22] | Vinberg, E.B.: Complexity of actions of reductive groups. Funktsional. Anal. i Prilozhen. 20, 1-13 (1986) · Zbl 0601.14038 · doi:10.1007/BF01077308 |
| [23] | Weist, T.: Tree modules. To appear in: Bulletin of the London Mathematical Society. arXiv:1011.1203 (2010) · Zbl 1219.16018 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
