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MINIMALLY TRANSITIVE FAMILIES OF SUBSPACES OF $\mathbb {C}^n$

Part of: Basic linear algebra

Published online by Cambridge University Press:  20 January 2022

W. E. LONGSTAFF Open the ORCID record for W. E. LONGSTAFF [Opens in a new window]
W. E. LONGSTAFF*
Affiliation:
11 Tussock Crescent, Elanora, Queensland 4221, Australia
*
e-mail: bill.longstaff@alumni.utoronto.ca
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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.

Keywords

transitiveminimalmedial

MSC classification

Primary: 15A30: Algebraic systems of matrices
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 1 , August 2022 , pp. 70 - 82
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

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