A Log-Log Saving for Matrix-Algebra Length and Terseness

2026/07/18 by Florian Ito Sprung
Mathematics · #math.CO #math.RA

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Abstract

Let ℓ(\Mat_n(F)) denote the length of the full matrix algebra for a field F, i.e. the largest of the least word length needed to span \Mat_n(F), over all generating sets S of \Mat_n(F). Šitov proved the general estimate ℓ(\Mat_n(F)) ≤ 2nlog₂ n+4n-4. The purpose of this paper is to obtain a log-log saving, and prove that for every n>1, ℓ(\Mat_n(F)) ≤ 2nlog₂ n-2nlog₂log₂ n+5n. A theorem of Specht gives a word-criterion for unitary similarity of complex n× n matrices. The trace argument of Freedman--Gupta--Guralnick, as used by Pappacena, shows that any upper bound on ℓ(\Mat_n(F)) can be used to bound the terseness τ(n), i.e. the least upper bound for the length of words needed in Specht's theorem. Thus, for n> 1, τ(n)≤ 4nlog₂ n-4nlog₂log₂ n+10n+1.

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