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

DOI:
https://doi.org/10.1090/S0002-9947-1982-0675077-2

MathSciNet review:
675077

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Abstract: If is a finite-dimensional vector space over or and , the set of -dimensional -invariant subspaces is a compact subvariety of the Grassmann manifold , but it need not be a Schubert variety. We study the topology of . We reduce to the case where is nilpotent. In this case we prove that is connected but need not be a manifold. However, the subset of consisting of those subspaces with a fixed cyclic structure is a regular submanifold of .

**[1]**W. M. Boothby,*An introduction to differentiable manifolds and Riemannian geometry*, Academic Press, New York, 1975. MR**0426007 (54:13956)****[2]**T. Brylawski,*The lattice of integer partitions*, Discrete Math.**6**(1973), 201-219. MR**0325405 (48:3752)****[3]**F. R. Gantmacher,*The theory of matrices*, Vol. I, Chelsea, New York, 1959.**[4]**P. Griffiths and J. Harris,*Principles of algebraic geometry*, Wiley, New York, 1978. MR**507725 (80b:14001)****[5]**J. E. Humphreys,*Linear algebraic groups*, Springer-Verlag, New York, 1975. MR**0396773 (53:633)****[6]**S. L. Kleiman,*Problem 15, rigorous foundations of Schubert's enumerative calculus*, Proc. Sympos. Pure Math., vol. 28, Math. Developments Arising from Hilbert Problems, Amer. Math. Soc., Providence, R. I., 1976. MR**0429938 (55:2946)****[7]**M. A. Shayman,*Varieties of invariant subspaces and the algebraic Riccati equation*, Ph. D. thesis, Harvard University, 1980.**[8]**J. C. Willems,*Least squares stationary optimal control and the algebraic Riccati equation*, IEEE Trans. Automat. Control**16**(1971), 621-634. MR**0308890 (46:8002)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0675077-2

Keywords:
Invariant subspace,
Grassmann manifold,
Schubert variety

Article copyright:
© Copyright 1982
American Mathematical Society