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

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

Problem Statement
The paper addresses a classical problem in linear and associative algebra: determining the minimal possible dimension of a maximal commutative subalgebra $A \subseteq M_n(K)$, where $K$ is an algebraically closed field and $M_n(K)$ is the algebra of $n \times n$ matrices. While large-dimensional examples are well-understood, the existence of maximal commutative algebras with dimensions strictly smaller than $n$ (referred to as "Courter-like" algebras) is a subtle and less understood phenomenon.

Historically, the algebra of diagonal matrices provides a maximal commutative subalgebra of dimension $n$. In 1961, Courter constructed an example in $M_{14}(K)$ with dimension 13, proving that $\dim A < n$ is possible. Prior to this work, no examples were known for $n < 14$, and the general lower bound established by Laffey, $\dim A > (2n)^{2/3} - 1$, was insufficient to rule out smaller exceptional cases. The specific gap in knowledge concerned the range $7 \le n \le 13$, where neither a classification nor a proof of the inequality $\dim A \ge n$ existed.

Methodology
The authors employ a structural approach based on the normal form theorem of Suprunenko and Tyshkevich. The methodology proceeds through the following steps:

1. Reduction to Local Algebras: Using a proposition by Suprunenko and Tyshkevich, the authors focus on local maximal commutative algebras (those where every element is either invertible or nilpotent). They establish that any maximal commutative algebra decomposes into a direct sum of local maximal commutative algebras.
2. Signature Analysis: Local maximal commutative algebras are conjugate to a block lower-triangular form defined by a signature $(n_1, n_2, n_3)$, where $n = n_1 + n_2 + n_3$. The algebra structure is determined by a socle block $\Sigma \cong M_{n_3 \times n_1}(K)$ and a radical layer $\Lambda$ generated by nilpotent elements.
3. Centralizer Estimation: The authors derive a sharp lower bound for the dimension of the centralizer $C(A)$ for an algebra with a given signature. By imposing the condition of maximal commutativity ($C(A) = A$), they derive a formula relating the dimension of the radical generators ($d$) to the signature components.
4. Optimization of Signatures: The authors analyze the function $F(n_1, n_2, n_3)$ representing the lower bound of the dimension. They prove that for $n_1, n_3 \ge 2$, the minimal dimensions occur in the boundary family of signatures $(2, s, 2)$.
5. Stack Construction: To demonstrate that the derived lower bounds are attainable, the authors introduce a "stack construction." This operation combines two algebras of signature $(2, s, 2)$ into a larger algebra of signature $(2, s_1+s_2, 2)$ while preserving maximal commutativity and exhibiting a specific dimension-saving property ($\dim(A_1 \star A_2) = \dim A_1 + \dim A_2 - 5$).

Key Contributions and Results

• Sharp Lower Bounds: The paper establishes an exact lower bound for the dimension of any maximal commutative subalgebra $A \subseteq M_n(K)$:
  $$\dim A \ge \begin{cases} n & \text{if } n \le 8 \\ 5 + \lceil \frac{4(n-4)}{5} \rceil & \text{if } n \ge 9 \end{cases}$$
  (Note: The paper extends the $n \le 8$ result to $n \le 13$ in the abstract and discussion, confirming no Courter-like algebras exist in this range).

• Resolution of the $n \le 13$ Case: The authors prove that for all $n \le 13$, any maximal commutative subalgebra must satisfy $\dim A \ge n$. Consequently, no Courter-like algebras (where $\dim A < n$) exist in $M_n(K)$ for $n \le 13$. This refines Laffey's estimate and closes the gap for small $n$.

• Identification of the First Exceptional Case: The paper confirms that $n=14$ is the first dimension where $\dim A < n$ is possible. Courter's classical example in $M_{14}(K)$ (signature $(2, 10, 2)$ with dimension 13) is shown to be the first possible exceptional case and it attains the optimal bound derived in the paper.

• Explicit Infinite Families: The authors construct explicit infinite families of maximal commutative subalgebras that attain the lower bound for all $n \ge 14$. This is achieved by defining three "building blocks":

  1. A seed algebra $E \subseteq M_9(K)$ of signature $(2, 5, 2)$ with dimension 9.
  2. Courter's algebra $C \subseteq M_{14}(K)$ of signature $(2, 10, 2)$ with dimension 13.
  3. A diagonal brick $D \subseteq M_5(K)$ of signature $(2, 1, 2)$ with dimension 6.
     By stacking these blocks (specifically $E$ and $D$), the authors generate algebras $A_n$ for any $n \ge 14$ such that $\dim A_n = 5 + \lceil \frac{4(n-4)}{5} \rceil$.

Significance and Claims
The paper claims to refine the classical estimates of Laffey by determining the exact lower bounds for maximal commutative subalgebras. Its primary significance lies in:

1. Ruling out small exceptions: Proving definitively that Courter-like algebras do not exist for $n \le 13$.
2. Characterizing the first exception: Identifying Courter's $M_{14}$ example not as an isolated anomaly, but as the first member of an infinite, sharp family of minimal maximal commutative matrix algebras.
3. Constructive Proof: Providing a constructive method (the stack construction) to generate minimal examples in every dimension $n \ge 14$, thereby showing the bound is tight.

The authors conclude that the global extremal problem for minimal maximal commutative subalgebras is essentially solved, provided one understands the local building blocks. The remaining difficulty is described as "entirely local," focusing on the classification of local maximal commutative algebras with small dimensions relative to the matrix size.