On the length of a class of maximal commutative subalgebras
Chengjie Wang
TL;DR
We address the problem of determining the length of maximal commutative subalgebras of $M_n(F)$ and extend the landscape of known constructions. The authors introduce the class $B_{k,m,l}$, generated by $E_n$, $B_1=E_{m,m+1}+\\cdots+E_{m+k,m+k+1}$, $B_2=E_{l,l+1}+\\cdots+E_{l+k,l+k+1}$, and certain matrix units $E_{i,j}$, and prove it is a maximal commutative subalgebra. The main result is $\ell(B_{k,m,l})=k+1$ and the radical has nilpotency index $N=k+2$, with the proof combining an upper bound from general theory with a carefully constructed generating set to realize the lower bound. Two explicit examples demonstrate that this construction does not straightforwardly generalize a prior proposition, illustrating delicate parameter dependence. Overall, the work enhances the understanding of the structure and length of maximal commutative subalgebras in matrix algebras.
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.