On the sum of the dimension of a matrix subalgebra and its centralizer

TL;DR

This paper investigates the sum of dimensions of two commuting subalgebras of matrix algebra over a field, establishing an upper bound and characterizing cases of equality.

Contribution

It provides a new upper bound on the sum of dimensions for commuting subalgebras containing non-scalar matrices and characterizes the equality cases.

Findings

Dimension sum bound: (n-1)^2+3

Complete description of equality cases

Applicable to matrix subalgebra classification

Abstract

When $\mathbb{K}$ is a field, and $\mathcal{A}$ and $\mathcal{B}$ denote commuting subspaces of $\text{M}_n(\K)$ each of which contains a non-scalar matrix, we prove that $\dim \mathcal{A} +\dim \mathcal{B} \leq (n-1)^2+3$. We also give a complete description of the cases when equality holds.