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Small Maximal Spaces of Non‐Invertible Matrices
Author(s) -
Draisma Jan
Publication year - 2006
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609306018741
Subject(s) - mathematics , rank (graph theory) , invertible matrix , dimension (graph theory) , vector space , space (punctuation) , matrix (chemical analysis) , field (mathematics) , pure mathematics , combinatorics , algebra over a field , discrete mathematics , philosophy , linguistics , materials science , composite material
The rank of a vector space A of n × n ‐matrices is by definition the maximal rank of an element of A . The space A is called rank‐critical if any matrix space that properly contains 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 × 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. 2000 Mathematics Subject Classification 15A30, 17B10, 20G05.