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A swiss cheese theorem for linear operators with two invariant subspaces


Authors: Audrey Moore and Markus Schmidmeier
Journal: Proc. Amer. Math. Soc. 143 (2015), 5097-5111
MSC (2010): Primary 16G20, 47A15
DOI: https://doi.org/10.1090/proc12754
Published electronically: May 20, 2015
MathSciNet review: 3411129
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Abstract: We study systems $ (V,T,U_1,U_2)$ consisting of a finite dimensional vector space $ V$, a nilpotent $ k$-linear operator $ T:V\to V$ and two $ T$-invariant subspaces $ U_1\subset U_2\subset V$. Let $ \mathcal S(n)$ be the category of such systems where the operator $ T$ acts with nilpotency index at most $ n$. We determine the dimension types $ (\dim U_1, \dim U_2/U_1, \dim V/U_2)$ of indecomposable systems in $ \mathcal S(n)$ for $ n\leq 4$. It turns out that in the case where $ n=4$ there are infinitely many such triples $ (x,y,z)$, they all lie in the cylinder given by $ \vert x-y\vert,\vert y-z\vert$,
$ \vert z-x\vert\leq 4$. But not each dimension type in the cylinder can be realized by an indecomposable system. In particular, there are holes in the cylinder. Namely, no triple in $ (x,y,z)\in (3,1,3)+\mathbb{N}(2,2,2)$ can be realized, while each neighbor $ (x\pm 1,y,z), (x,y\pm 1,z),(x,y,z\pm 1)$ can. Compare this with Bongartz' No-Gap Theorem, which states that for an associative algebra $ A$ over an algebraically closed field, there is no gap in the lengths of the indecomposable $ A$-modules of finite dimension.


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  • [1] Garrett Birkhoff, Subgroups of abelian groups, Proc. London Math. Soc. S2-38, no. 1, 385. MR 1576323, https://doi.org/10.1112/plms/s2-38.1.385
  • [2] Klaus Bongartz, Indecomposables live in all smaller lengths, Represent. Theory 17 (2013), 199-225. MR 3038490, https://doi.org/10.1090/S1088-4165-2013-00429-6
  • [3] Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314 (83g:17001)
  • [4] Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124 (89e:16035)
  • [5] Dirk Kussin, Helmut Lenzing, and Hagen Meltzer, Nilpotent operators and weighted projective lines, J. Reine Angew. Math. 685 (2013), 33-71. MR 3181563
  • [6] Huibert Kwakernaak and Raphael Sivan, Linear optimal control systems, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1972. MR 0406607 (53 #10394)
  • [7] Audrey Moore, Auslander-Reiten theory for systems of submodule embeddings, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)-Florida Atlantic University. MR 2713354
  • [8] Audrey Moore, The Auslander and Ringel-Tachikawa theorem for submodule embeddings, Comm. Algebra 38 (2010), no. 10, 3805-3820. MR 2760692 (2011k:16040), https://doi.org/10.1080/00927870903286843
  • [9] Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589 (87f:16027)
  • [10] Claus Michael Ringel and Markus Schmidmeier, Invariant subspaces of nilpotent linear operators. I, J. Reine Angew. Math. 614 (2008), 1-52. MR 2376281 (2009d:16016), https://doi.org/10.1515/CRELLE.2008.001
  • [11] Claus Michael Ringel, Indecomposables live in all smaller lengths, Bull. Lond. Math. Soc. 43 (2011), no. 4, 655-660. MR 2820151 (2012e:16015), https://doi.org/10.1112/blms/bdq128
  • [12] Daniel Simson, Linear representations of partially ordered sets and vector space categories, Algebra, Logic and Applications, vol. 4, Gordon and Breach Science Publishers, Montreux, 1992. MR 1241646 (95g:16013)
  • [13] 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 (2009b:16040), https://doi.org/10.1016/j.jalgebra.2007.01.029
  • [14] Bao-Lin Xiong, Pu Zhang, and Yue-Hui Zhang, Auslander-Reiten translations in monomorphism categories, Forum Math. 26 (2014), no. 3, 863-912. MR 3200353, https://doi.org/10.1515/forum-2011-0003

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Additional Information

Audrey Moore
Affiliation: Department of Mathematical Sciences, Delaware State University, 1200 N DuPont Highway, Dover, Delaware 19901
Address at time of publication: Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, Florida 33431
Email: audreydoughty@yahoo.com

Markus Schmidmeier
Affiliation: Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, Florida 33431
Email: markus@math.fau.edu

DOI: https://doi.org/10.1090/proc12754
Keywords: Linear operators, invariant subspaces, No-Gap Theorem, tubular algebras
Received by editor(s): September 22, 2014
Published electronically: May 20, 2015
Additional Notes: This research was partially supported by a Travel and Collaboration Grant from the Simons Foundation (Grant number 245848 to the second-named author).
Communicated by: Harm Derksen
Article copyright: © Copyright 2015 American Mathematical Society

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