On the length of a class of maximal commutative subalgebras

TL;DR

This paper investigates the lengths of a specific class of maximal commutative subalgebras in matrix algebras, providing explicit calculations and examples to deepen understanding of their structure.

Contribution

It introduces a new class of maximal commutative subalgebras and computes their lengths, highlighting complexities beyond straightforward generalizations.

Findings

Computed lengths of the subalgebras $\

,

and $\

Abstract

A maximal commutative subalgebra is a substructure in algebra with the greatest commutative property. By studying the lengths of maximal commutative subalgebras, one can more clearly characterize the structure of commutative subalgebras in the full matrix algebra $\mathrm{M}_n({\mathbb{F}})$. Inspired by \cite[Proposition~4.12]{markova2013}, this paper identifies a class of maximal commutative subalgebras $\mathcal{B}_{k,m,l}$ and computes their lengths. Finally, we present two concrete examples to show that it is not a straightforward generalization.