Operations on arc diagrams and degenerations for invariant subspaces of linear operators

Kosakowska, Justyna; Schmidmeier, Markus

doi:10.1090/S0002-9947-2014-06206-5

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