On affine spaces of alternating matrices with constant rank

2023/07/19 by Pazzis, Clément de Seguins · 1 citation
#15A03 #15A30 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2307.10347

Abstract

Let \mathbbF be a field, and n ≥ r>0 be integers, with r even. Denote by A_n(\mathbbF) the space of all n-by-n alternating matrices with entries in \mathbbF. We consider the problem of determining the greatest possible dimension for an affine subspace of A_n(\mathbbF) in which every matrix has rank equal to r (or rank at least r). Recently Rubei has solved this problem over the field of real numbers. We extend her result to all fields with large enough cardinality. Provided that n ≥ r+3 and |\mathbbF|≥ min(r-1,(r)/(2)+2), we also determine the affine subspaces of rank r matrices in A_n(\mathbbF) that have the greatest possible dimension, and we point to difficulties for the corresponding problem in the case n≤ r+2.

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