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arxiv: 2605.01387 · v2 · pith:A52QHNT6new · submitted 2026-05-02 · 🧮 math.RA · math.AC

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

Ma{\l}gorzata Nowak-K\k{e}pczyk This is my paper

Pith reviewed 2026-05-21 00:52 UTC · model grok-4.3

classification 🧮 math.RA math.AC
keywords maximal commutative subalgebrasmatrix algebrasdimension boundsCourter examplestack constructionLaffey boundalgebraically closed field
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The pith

Maximal commutative subalgebras in M_n(K) have dimension at least n for all n ≤ 13, with Courter's example attaining the optimal bound at n=14.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines sharp lower bounds for the dimension of maximal commutative subalgebras A in the matrix algebra M_n(K) over an algebraically closed field K. It proves that this dimension is at least n whenever n is 13 or less, eliminating the possibility of Courter-like algebras with smaller dimension in this range. The analysis shows that the known example by Courter for n=14 is the smallest exception to the n bound and that it achieves the best possible lower bound. A stack construction is introduced to build explicit infinite families of maximal commutative subalgebras that attain this optimal dimension for every n at least 14.

Core claim

We prove that dim A >= n for all n <= 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 >= 14.

What carries the argument

The stack construction for building infinite families of maximal commutative subalgebras that attain the minimal dimension bound.

If this is right

  • For n ≤ 13, every maximal commutative subalgebra of M_n(K) has dimension at least n.
  • Courter's example provides a maximal commutative subalgebra in M_14(K) of minimal possible dimension.
  • The stack construction produces maximal commutative subalgebras of minimal dimension for all n ≥ 14.
  • These results refine the classical lower bound given by Laffey to sharp values in each range.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The pattern suggests that the minimal dimension may follow a linear function of n for large n.
  • Similar constructions might apply to maximal commutative subalgebras in other types of algebras.
  • Questions about the existence of such algebras over finite fields or reals could be explored using different techniques.

Load-bearing premise

The field K is algebraically closed, which is needed to guarantee eigenvalues and Jordan forms for proving the dimension lower bounds.

What would settle it

Discovery of a maximal commutative subalgebra A in M_n(K) for some n ≤ 13 with dim A < n would falsify the claim, or failure of the stack construction to yield maximal algebras for some large n.

Figures

Figures reproduced from arXiv: 2605.01387 by Ma{\l}gorzata Nowak-K\k{e}pczyk.

Figure 1
Figure 1. Figure 1: Comparison of Laffey’s classical lower bound, the Courter-type bound [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Comparison of the signature-free local lower bounds [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
read the original 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 $n \ge 14$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper determines sharp lower bounds on the dimension of maximal commutative subalgebras A of M_n(K) for algebraically closed K. It proves dim A ≥ n for all n ≤ 13 (hence no Courter-type examples exist in this range), shows that Courter's n=14 example is the smallest exceptional case and attains the optimal bound, and introduces an explicit stack construction yielding infinite families of maximal commutative subalgebras attaining the bound for every n ≥ 14.

Significance. If the proofs are correct, the work sharpens Laffey's classical estimate with precise thresholds and provides the first explicit infinite families attaining the minimal dimension via the stack construction. The direct verification that the centralizer equals the algebra itself in the construction, together with the case analysis for small n, strengthens the result and supplies falsifiable predictions for the minimal dimension as a function of n.

minor comments (2)
  1. The definition of the stack construction in the main theorem for n ≥ 14 would benefit from an explicit small-n example (e.g., n=14 or n=15) showing the block sizes and the verification that the centralizer equals the algebra.
  2. A brief table or list summarizing the proven minimal dimensions for 1 ≤ n ≤ 20 would improve readability and make the transition from the n ≤ 13 case analysis to the n ≥ 14 families clearer.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. The referee summary correctly captures the main results, including the proof that dim A >= n for n <= 13, the optimality of Courter's example at n=14, and the stack construction for infinite families when n >= 14.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via direct proofs and explicit construction

full rationale

The paper establishes lower bounds dim A >= n for n <= 13 through case analysis and extension arguments on maximal commutative subalgebras, identifies Courter's M_14 example as the first exception attaining the bound, and introduces an explicit stack construction yielding infinite families for n >= 14 whose maximality is verified directly by showing the centralizer equals the algebra. These steps rely on standard facts about Jordan canonical forms and eigenvalues over algebraically closed fields, invoked in the expected places without presupposing the target dimension bounds. No equations or claims reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the algebraic-closure hypothesis is a standard external assumption rather than a hidden circular premise. The work is internally consistent with classical matrix algebra theory and does not rename known results or smuggle ansatzes via prior self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard domain assumption that the scalar field is algebraically closed; no free parameters or newly invented entities are indicated in the abstract.

axioms (1)
  • domain assumption K is an algebraically closed field
    Invoked at the beginning of the abstract to set up the matrix algebra M_n(K) and to support the subsequent dimension arguments.

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · 1 internal anchor

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