On the minimal dimension of maximal commutative subalgebras of $M_6(k)$

TL;DR

This paper proves that for 6x6 matrix algebras over an algebraically closed field, every maximal commutative subalgebra has dimension at least 6, using analysis of local algebras and centralizers.

Contribution

It establishes a lower bound of 6 on the dimension of maximal commutative subalgebras of M_6(k), filling a gap in known examples for smaller n.

Findings

Maximal commutative subalgebras of M_6(k) have dimension at least 6.

Examples with smaller dimension are known only for n ≥ 14.

The proof involves analysis of local algebras and module structures.

Abstract

We study the minimal dimension of maximal commutative subalgebras of the matrix algebra $M_n(k)$ over an algebraically closed field. While examples with dimension strictly smaller than n are known for $n \geq 14$, no such examples are known in smaller dimensions. In this paper, we show that for n = 6 every maximal commutative subalgebra $A\subset M_6(k)$ satisfies $\dim A \geq 6$. The proof is based on a detailed analysis of local algebras and their module structure, combined with explicit estimates of the dimension of the centralizer.