Minimal Dimensions of Maximal Commutative Matrix Algebras and Sharp Courter-Type Bounds
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914574072020992 |
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| author | Nowak-Kępczyk, Małgorzata |
| author_facet | Nowak-Kępczyk, Małgorzata |
| contents | 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 $n \ge 14$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01387 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimal Dimensions of Maximal Commutative Matrix Algebras and Sharp Courter-Type Bounds Nowak-Kępczyk, Małgorzata Rings and Algebras Commutative Algebra 15A30, 15A27, 16S50 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 $n \ge 14$. |
| title | Minimal Dimensions of Maximal Commutative Matrix Algebras and Sharp Courter-Type Bounds |
| topic | Rings and Algebras Commutative Algebra 15A30, 15A27, 16S50 |
| url | https://arxiv.org/abs/2605.01387 |