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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
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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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

Markus Schmidmeier
Affiliation: Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, Florida 33431

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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