An improved bound for the lengths of matrix algebras
Yaroslav Nikolaevich Shitov · Algebra & Number Theory · 2019
An improved bound for the lengths of matrix algebras Yaroslav ShitovLet S be a set of n × n matrices over a field .ކWe show that the -ކlinear span of the words in S of length at most 2n log 2 n + 4nis the full -ކalgebra generated by S. This improves on the n 2 3 + 2 3 bound by Paz (1984) and an O(n 3/2 ) bound of Pappacena (1997).Let S be a subset of a finite-dimensional associative algebra A over a field .ކAn element a ∈ A is said to be a word of length k in S if there are a 1 , . . ., a k ∈ S such that a = a 1 • • • a k .We denote the set of all such words by S k , and we write ކS k for the -ކlinear span of S k .Similarly, ކS k will stand for the -ކlinear span of all the words in S that have length at most k.Definition 1.The length ℓ(S) is the smallest integer k for which ކS k is the full subalgebra generated by S. We also define ℓ(A) as the maximum value of ℓ(S), where S runs over all subsets of A that generate A as an -ކalgebra.In our paper, we study the length of Mat n ,)ކ( the set of n × n matrices viewed as an algebra over . ކ A. Paz [1984] proved that ℓ(S) n 2 3 + 2 3 for all S ⊂ Mat n )ކ( and proposed the following appealing conjecture.Conjecture 2. For all S ⊂ Mat n ,)ކ( one has ℓ(S) 2n -2.As shown by T. Laffey [1986, page 131], the upper bound in Conjecture 2 should be sharp.This conjecture is known to hold if the size of matrices is at most four [Paz 1984] or if ކS contains a nonderogatory matrix [Guterman et al. 2018].However, the best known general upper bounds on the lengths of matrix subsets are quite far from the one prescribed by Conjecture 2. It was only in 1997 when a subquadratic estimate was obtained: C. Pappacena proved an O(n 3/2 ) upper bound on the length of Mat n ,)ކ( but no further improvements have been made since then [Guterman et al. 2018;Lambrou and Longstaff 2009;Longstaff et al. 2006].The main result of this paper is a much stronger O(n log n) upper bound on the length of Mat n .)ކ( Theorem 3.For all S ⊂ Mat n ,)ކ( we have ℓ(S) 2n log 2 n + 4n -4.