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

Abstract

Let $\ell(\Mat_n(F))$ 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 $$ \ell(\Mat_n(F)) \leq 2n\log_2 n+4n-4. $$ The purpose of this paper is to obtain a log-log saving, and prove that for every $n>1$, $$ \ell(\Mat_n(F)) \leq 2n\log_2 n-2n\log_2\log_2 n+5n. $$ A theorem of Specht gives a word-criterion for unitary similarity of complex $n\times n$ matrices. The trace argument of Freedman--Gupta--Guralnick, as used by Pappacena, shows that any upper bound on $\ell(\Mat_n(F))$ can be used to bound the \emph{terseness} $\tau (n)$, i.e. the least upper bound for the length of words needed in Specht's theorem. Thus, for $n>1$, $$\tau (n)\le 4n{\log }_{2}n-4n{\log }_2{\log }_{2}n+10n+1.$$

7 pages+ references. Incorporated several improvements, and added a new reference. If you have any comments, please let me know