Abstract
We explore the relationship between local and global reducibility of spaces of nilpotent matrices. By local reducibility we mean that small subspaces of a given irreducible linear space L⊆Mn(C) are reducible. One of our main results is that for certain integers m depending on n there is an (m+1)-dimensional space L which is irreducible, but every one of its m-dimensional subspaces is, not just reducible, but simultaneously triangularizable.
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