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SMALL MAXIMAL SPACES OF NON-INVERTIBLE MATRICES

Published online by Cambridge University Press:  20 September 2006

JAN DRAISMA
Affiliation:
Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, [email protected]
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Abstract

The rank of a vector space $\mathcal{A}$ of $n\times n$-matrices is by definition the maximal rank of an element of $\mathcal{A}$. The space $\mathcal{A}$ is called rank-critical if any matrix space that properly contains $\mathcal{A}$ has a strictly higher rank. This paper exhibits a sufficient condition for rank-criticality, which is then used to prove that the images of certain Lie algebra representations are rank-critical. A rather counter-intuitive consequence, and the main novelty in this paper, is that for infinitely many $n$, there exists an eight-dimensional rank-critical space of $n \times n$-matrices having generic rank $n-1$, or, in other words: an eight-dimensional maximal space of non-invertible matrices. This settles the question, posed by Fillmore, Laurie, and Radjavi in 1985, of whether such a maximal space can have dimension smaller than $n$. Another consequence is that the image of the adjoint representation of any semisimple Lie algebra is rank-critical; in both results, the ground field is assumed to have characteristic zero.

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Papers
Copyright
© The London Mathematical Society 2006

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