Minimal Dimensions of Maximal Commutative Matrix Algebras and Sharp Courter-Type Bounds

TL;DR

This paper establishes sharp lower bounds for the dimensions of maximal commutative subalgebras in matrix algebras, refining classical estimates and identifying the first exceptional case at dimension 14.

Contribution

It provides the first sharp bounds for all dimensions, proves no Courter-like algebras exist for n ≤ 13, and constructs infinite families of maximal subalgebras attaining these bounds.

Findings

No Courter-like algebras for n ≤ 13.

Courter's example in M_{14}(K) is the first exceptional case.

Explicit infinite families of maximal subalgebras attain the bound for all n ≥ 14.

Abstract

Let $K$ be an algebraically closed field and let $M_n(K)$ denote the algebra of $n\times n$ matrices over $K$. A classical problem asks for the minimal possible dimension of a maximal commutative subalgebra $A \subseteq M_n(K)$. We determine sharp lower bounds for maximal commutative subalgebras of $M_n(K)$, refining the classical estimate of Laffey. In particular, we prove that $\dim A \ge n$ for all $n \le 13$, so no Courter-like algebras exist in this range. Moreover, we show that Courter's example in $M_{14}(K)$ is the first possible exceptional case and already attains the optimal bound. Finally, we introduce a stack construction and obtain explicit infinite families of maximal commutative subalgebras attaining the bound for all $n \ge 14$.