Operations on arc diagrams and degenerations for invariant subspaces of linear operators
HTML articles powered by AMS MathViewer
- by Justyna Kosakowska and Markus Schmidmeier PDF
- Trans. Amer. Math. Soc. 367 (2015), 5475-5505 Request permission
Abstract:
We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are algebraic groups acting on these varieties, and we give dimensions of orbits of these actions. Moreover, a combinatorial characterization of the partial order given by degenerations is described.References
- Garrett Birkhoff, Subgroups of Abelian Groups, Proc. London Math. Soc. (2) 38 (1935), 385–401. MR 1576323, DOI 10.1112/plms/s2-38.1.385
- Klaus Bongartz, On degenerations and extensions of finite-dimensional modules, Adv. Math. 121 (1996), no. 2, 245–287. MR 1402728, DOI 10.1006/aima.1996.0053
- Klaus Bongartz, Degenerations for representations of tame quivers, Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 5, 647–668. MR 1341664
- Christian U. Jensen and Helmut Lenzing, Model-theoretic algebra with particular emphasis on fields, rings, modules, Algebra, Logic and Applications, vol. 2, Gordon and Breach Science Publishers, New York, 1989. MR 1057608
- Stanisław Kasjan, Representation-directed algebras form an open scheme, Colloq. Math. 93 (2002), no. 2, 237–250. MR 1930801, DOI 10.4064/cm93-2-3
- Stanisław Kasjan and Justyna Kosakowska, On Lie algebras associated with representation-directed algebras, J. Pure Appl. Algebra 214 (2010), no. 5, 678–688. MR 2577675, DOI 10.1016/j.jpaa.2009.07.012
- T. Klein, The multiplication of Schur-functions and extensions of $p$-modules, J. London Math. Soc. 43 (1968), 280–284. MR 228481, DOI 10.1112/jlms/s1-43.1.280
- T. Klein, The Hall polynomial, J. Algebra 12 (1969), 61–78. MR 236260, DOI 10.1016/0021-8693(69)90017-9
- Justyna Kosakowska, Degenerations in a class of matrix varieties and prinjective modules, J. Algebra 263 (2003), no. 2, 262–277. MR 1978651, DOI 10.1016/S0021-8693(03)00173-X
- H. Kraft and Ch. Riedtmann, Geometry of representations of quivers, Representations of algebras (Durham, 1985) London Math. Soc. Lecture Note Ser., vol. 116, Cambridge Univ. Press, Cambridge, 1986, pp. 109–145. MR 897322
- Serge Lang and André Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827. MR 65218, DOI 10.2307/2372655
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
- Christine Riedtmann, Degenerations for representations of quivers with relations, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 275–301. MR 868301
- Claus Michael Ringel, Hall algebras, Topics in algebra, Part 1 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 433–447. MR 1171248
- Claus Michael Ringel and Markus Schmidmeier, Invariant subspaces of nilpotent linear operators. I, J. Reine Angew. Math. 614 (2008), 1–52. MR 2376281, DOI 10.1515/CRELLE.2008.001
- Markus Schmidmeier, Bounded submodules of modules, J. Pure Appl. Algebra 203 (2005), no. 1-3, 45–82. MR 2176651, DOI 10.1016/j.jpaa.2005.02.003
- Markus Schmidmeier, Hall polynomials via automorphisms of short exact sequences, Algebr. Represent. Theory 15 (2012), no. 3, 449–481. MR 2912467, DOI 10.1007/s10468-010-9250-6
- Daniel Simson, Representation types of the category of subprojective representations of a finite poset over $K[t]/(t^m)$ and a solution of a Birkhoff type problem, J. Algebra 311 (2007), no. 1, 1–30. MR 2309875, DOI 10.1016/j.jalgebra.2007.01.029
- Grzegorz Zwara, Degenerations for representations of extended Dynkin quivers, Comment. Math. Helv. 73 (1998), no. 1, 71–88. MR 1610587, DOI 10.1007/s000140050046
Additional Information
- Justyna Kosakowska
- Affiliation: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
- MR Author ID: 633460
- Email: justus@mat.umk.pl
- Markus Schmidmeier
- Affiliation: Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, Florida 33431
- MR Author ID: 618925
- ORCID: 0000-0003-3365-6666
- Email: markus@math.fau.edu
- Received by editor(s): February 13, 2012
- Received by editor(s) in revised form: August 16, 2012, and June 4, 2013
- Published electronically: December 24, 2014
- Additional Notes: The first named author was partially supported by Research Grant No. DEC-2011/02/A/ST1/00216 of the Polish National Science Center
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 5475-5505
- MSC (2010): Primary 14L30, 16G20; Secondary 16G70, 05C85, 47A15
- DOI: https://doi.org/10.1090/S0002-9947-2014-06206-5
- MathSciNet review: 3347180
Dedicated: Dedicated to Professor Daniel Simson on the occasion of his 70th birthday