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MINIMALLY TRANSITIVE FAMILIES OF SUBSPACES OF $\mathbb {C}^n$
Published online by Cambridge University Press: 20 January 2022
Abstract
Every transitive family of subspaces of a vector space of finite dimension $n\ge 2$ over a field $\mathbb {F}$ contains a subfamily which is transitive but has no proper transitive subfamily. Such a subfamily is called minimally transitive. Each has at most $n^2-n+1$ elements. On ${{\mathbb {C}}}^n, n\ge 3$ , a minimally transitive family of subspaces has at least four elements and a minimally transitive family of one-dimensional subspaces has $\tau $ elements where $n+1\le \tau \le 2n-2$ . We show how a minimally transitive family of one-dimensional subspaces arises when it consists of the subspaces spanned by the standard basis vectors together with those spanned by $0$ – $1$ vectors. On a space of dimension four, the set of nontrivial elements of a medial subspace lattice has five elements if it is minimally transitive. On spaces of dimension $12$ or more, the set of nontrivial elements of a medial subspace lattice can have six or more elements and be minimally transitive.
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- © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.