Matrix algebras up to size $13x13$ are mathematically forbidden from having exceptional minimal dimensions, setting a new sharp boundary at $14x14$.
This research settles a classical problem in linear algebra regarding the size of commutative subalgebras. It proves that for any matrix size up to 13, the minimal dimension must be at least the size of the matrix itself. This identifies the number 14 as the first possible place where the rules of linear algebra might break and allow for weird, smaller structures. It provides a definitive map for where these exceptional algebras can and cannot exist. This is a foundational result that will guide future research in abstract math and theoretical physics.
Minimal Dimensions of Maximal Commutative Matrix Algebras and Sharp Courter-Type Bounds
arXiv · 2605.01387
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 e