Small Maximal Spaces of Non-Invertible Matrices

The rank of a vector space A of n x n-matrices is by definitionthe maximal rank of an element of A. The space A is called rank-criticalif any matrix space that properly contains A has a strictlyhigher rank. This paper exhibits a sufficient condition forrank-criticality, which is then used to prove that the imagesof certain Lie algebra representations are rank-critical. Arather counter-intuitive consequence, and the main novelty inthis paper, is that for infinitely many n, there exists an eight-dimensionalrank-critical space of n x n-matrices having generic rank n– 1, or, in other words: an eight-dimensional maximalspace of non-invertible matrices. This settles the question,posed by Fillmore, Laurie, and Radjavi in 1985, of whether sucha maximal space can have dimension smaller than n. Another consequenceis that the image of the adjoint representation of any semisimpleLie algebra is rank-critical; in both results, the ground fieldis assumed to have characteristic zero. 2000 Mathematics SubjectClassification 15A30, 17B10, 20G05.