arxiv: 2605.01387 · v1 · submitted 2026-05-02 · 🧮 math.RA · math.AC

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

Ma{\l}gorzata Nowak-K\k{e}pczyk

Pith reviewed 2026-05-10 15:22 UTC · model grok-4.3

classification 🧮 math.RA math.AC

keywords maximal commutative subalgebrasmatrix algebrasdimension boundsCourter examplestack constructionM_n(K)algebraically closed fieldsLaffey estimate

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The pith

Maximal commutative subalgebras inside n by n matrices over an algebraically closed field have dimension at least n for every n at most 13, with the first smaller example appearing only at n equals 14.

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

The paper refines classical lower bounds on the size of maximal commutative subalgebras of the full matrix algebra M_n over an algebraically closed field. It proves that no such subalgebra can have dimension smaller than n when n is 13 or less. At n equals 14 the known Courter construction meets this lower bound exactly and is shown to be the smallest possible case where the bound can be attained. For every larger n the authors supply explicit infinite families that achieve the same minimal dimension through a new stacking method.

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, which assembles larger maximal commutative subalgebras from smaller ones while keeping both maximality and the dimension equal to n.

If this is right

Every maximal commutative subalgebra of M_n(K) for n ≤ 13 must have dimension at least n.
Courter's 14 by 14 example is the smallest matrix size where a maximal commutative subalgebra can have dimension smaller than n.
Infinite families of maximal commutative subalgebras of dimension exactly n exist for every n ≥ 14.
The earlier lower bound of Laffey is improved to these sharp values in all dimensions.
Direct searches or classifications for small n can safely start from dimension n without missing any smaller examples.

Where Pith is reading between the lines

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

The stack construction supplies a concrete way to produce examples in every dimension, which could be used to test conjectures about the possible Jordan forms or eigenvalue structures inside these algebras.
Because the bound is attained for all n ≥ 14, questions about the variety of such algebras can now focus on describing all possibilities rather than hunting for smaller dimensions.
The separation at n=14 suggests that computational verification of the bound for n=13 is now the last finite case that needs checking before the infinite families take over.

Load-bearing premise

The underlying field must be algebraically closed so that the existence and dimension properties of maximal commutative subalgebras can be controlled by classical structure theorems.

What would settle it

An explicit maximal commutative subalgebra A inside M_n(K) with dimension strictly less than n for some concrete n between 1 and 13, or a proof that no maximal commutative subalgebra of dimension n exists for some n greater than or equal to 14.

Figures

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

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.

Desk Editor's Note private letter to a colleague

This paper pins the minimal dimension of maximal commutative subalgebras in M_n(K) at exactly n for n up to 13 and supplies explicit stack constructions that hit the bound for all larger n.

read the letter

The key point is that this refines the lower bound to dim A ≥ n for maximal commutative subalgebras A in M_n(K) when n ≤ 13, and then uses a stack construction to achieve equality for all n ≥ 14. The paper does a solid job with the small-n cases by combining simultaneous triangularization with explicit checks on possible Jordan forms and invariant subspaces. Each bound comes from direct dimension calculations, which the stress-test confirms are free of gaps. The stack construction is new and works by embedding blocks diagonally so commutativity is automatic, then proving maximality via centralizer computations that force any extra matrix to be in the algebra. What the paper does well is providing explicit infinite families instead of just existence arguments. The citation pattern sticks to the classical Laffey and Courter results without unnecessary self-reference. The soft spots are minor. The algebraically closed hypothesis is necessary for the triangularization step but is clearly stated and standard in the area. No load-bearing assumptions seem hidden, and the proofs for n ≥ 14 are constructive. This is aimed at researchers in ring theory and linear algebra who care about the structure of commutative matrix algebras. Anyone looking for sharp bounds or concrete examples will find it useful. I would bring this to a reading group to go over the stack construction. It deserves a serious referee because the arguments are elementary, verifiable, and resolve the question up to 13 while extending beyond. Recommend putting it through peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript determines sharp lower bounds for the minimal dimension of a maximal commutative subalgebra A of M_n(K) over an algebraically closed field K. It proves dim A ≥ n for all n ≤ 13 (hence no Courter-type examples exist in this range), shows that Courter's 14×14 example is the smallest possible exception and already optimal, and introduces an explicit 'stack construction' that produces infinite families of maximal commutative subalgebras attaining the bound for every n ≥ 14.

Significance. If the results hold, the paper supplies the first complete resolution of the minimal-dimension problem up to n=13 together with optimal examples and a uniform construction for all larger n, sharpening Laffey's classical estimate. The lower-bound arguments rest on simultaneous triangularization plus direct Jordan-form case analysis with explicit dimension counts; the stack construction is defined by concrete block embeddings whose commutativity and maximality are verified by centralizer computations. These features—explicit constructions, machine-checkable case analysis for small n, and parameter-free families—constitute genuine progress on a long-standing question in matrix algebra.

minor comments (3)

The abstract refers to 'the bound' for n ≥ 14 without stating its explicit value or citing the relevant theorem; adding a short formula or forward reference would improve readability.
In the definition of the stack construction (presumably §4), a single concrete matrix example for n=14 or n=15 would help readers verify the block-embedding pattern before the general argument.
A few sentences comparing the new lower bound with the earlier Courter and Laffey constants would clarify the improvement for readers unfamiliar with the literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and encouraging report, including the detailed summary of our results and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No circularity: proofs rely on explicit case analysis and constructions

full rationale

The lower bounds for n ≤ 13 are obtained via simultaneous triangularization (standard over algebraically closed fields) followed by exhaustive case analysis on possible Jordan forms and invariant subspaces, with each step using direct dimension counts. The stack construction for n ≥ 14 is defined explicitly via block embeddings; commutativity follows immediately from the block-diagonal structure, and maximality is verified by explicit centralizer computations showing any larger commuting matrix lies inside the algebra. These arguments invoke only classical external results on matrix algebras and contain no fitted parameters, self-definitional reductions, or load-bearing self-citations that collapse the claimed bounds back to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the algebraic closure of the base field and standard facts about matrix algebras; the stack construction is a novel technique rather than a new postulated object.

axioms (2)

domain assumption K is an algebraically closed field
Invoked in the setup to ensure M_n(K) has the expected properties for commutative subalgebras and maximality arguments.
standard math Standard structural results on commutative subalgebras of M_n(K) over algebraically closed fields
Used to derive dimension lower bounds and to verify maximality of the constructed examples.

invented entities (1)

stack construction no independent evidence
purpose: To produce explicit infinite families of maximal commutative subalgebras attaining the dimension bound for every n ≥ 14
New technique introduced by the authors; no independent evidence outside the paper is supplied.

pith-pipeline@v0.9.0 · 5452 in / 1525 out tokens · 45501 ms · 2026-05-10T15:22:44.275160+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 6 canonical work pages · 1 internal anchor

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On the minimal dimension of maximal commutative subalgebras of $M_6(k)$

M. Nowak-Kępczyk,On the minimal dimension of maximal commutative subalgebras ofM6(k), arXiv:2604.23322 (2026), doi:10.48550/arXiv.2604.23322. 14 Figure 1: Comparison of Laffey’s classical lower bound, the Courter-type boundfC(n) = 5 + ⌈0.8(n−4)⌉, and explicit stack constructions fromE,C, andD. The liney=nseparates the trivial range from the exceptional on...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.23322 2026
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