The effect of assuming the identity as a generator on the length of the matrix algebra

TL;DR

This paper investigates how removing the assumption that the identity matrix is included in generating sets affects the minimal length needed to generate the full matrix algebra.

Contribution

It explores the impact of not assuming the identity matrix is part of the generating set on the algebra's generating length.

Findings

The length of generating sets can be affected by the inclusion or exclusion of the identity.

Removing the identity assumption changes the minimal generating length.

The paper provides insights into the structure of generating sets without the identity constraint.

Abstract

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices and let $\mathcal S$ be a generating set of $M_n(\mathbb{F})$ as an $\mathbb{F}$-algebra. The length of a finite generating set $\mathcal S$ of $M_n(\mathbb{F})$ is the smallest number $k$ such that words of length not greater than $k$ generate $M_n(\mathbb{F})$ as a vector space. Traditionally the identity matrix is assumed to be automatically included in all generating sets $\mathcal S$ and counted as a word of length $0$. In this paper we discuss how the problem changes if this assumption is removed.