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arXiv · 1210.4951

On Gerstenhaber's theorem for spaces of nilpotent matrices over a skew field

Clément de Seguins Pazzis
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Abstract

Let K be a skew field, and K_0 be a subfield of the central subfield of K such that K has finite dimension q over K_0. Let V be a K_0-linear subspace of n by n nilpotent matrices with entries in K. We show that the dimension of V is bounded above by q n(n-1)/2, and that equality occurs if and only if V is similar to the space of all n by n strictly upper-triangular matrices over K. This generalizes famous theorems of Gerstenhaber and Serezhkin, which cover the special case K=K_0.

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Clément de Seguins Pazzis. 2013-02-23. On Gerstenhaber's theorem for spaces of nilpotent matrices over a skew field. https://doi.org/10.1016/j.laa.2013.01.035

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