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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the variety of invariant subspaces of a finite-dimensional linear operator

Author: Mark A. Shayman
Journal: Trans. Amer. Math. Soc. 274 (1982), 721-747
MSC: Primary 15A04; Secondary 14M15
MathSciNet review: 675077
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Abstract: If $ V$ is a finite-dimensional vector space over $ \mathbf{R}$ or $ \mathbf{C}$ and $ A \in {\operatorname {Hom}}(V)$, the set $ {S_A}(k)$ of $ k$-dimensional $ A$-invariant subspaces is a compact subvariety of the Grassmann manifold $ {G^k}(V)$, but it need not be a Schubert variety. We study the topology of $ {S_A}(k)$. We reduce to the case where $ A$ is nilpotent. In this case we prove that $ {S_A}(k)$ is connected but need not be a manifold. However, the subset of $ {S_A}(k)$ consisting of those subspaces with a fixed cyclic structure is a regular submanifold of $ {G^k}(V)$.

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Keywords: Invariant subspace, Grassmann manifold, Schubert variety
Article copyright: © Copyright 1982 American Mathematical Society