Irredundant Generating Sets for Matrix Algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908298299572224 |
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| author | Blumenthal, Yonatan First, Uriya |
| author_facet | Blumenthal, Yonatan First, Uriya |
| contents | Let $F$ be a field. We show that the largest irredundant generating sets for the algebra of $n\times n $ matrices over $F$ have $2n-1$ elements when $n>1$. (A result of Laffey states that the answer is $2n-2$ when $n>2$, but its proof contains an error.) We further give a classification of the largest irredundant generating sets when $n\in\{2,3\}$ and $F$ is algebraically closed. We use this description to compute the dimension of the variety of $(2n-1)$-tuples of $n\times n$ matrices which form an irredundant generating set when $n\in\{2,3\}$, and draw some consequences to Zariski-locally redundant generation of Azumaya algebras. In the course of proving the classification, we also determine the largest sets $S$ of subspaces of $F^3$ with the property that every $V\in S$ admits a matrix stabilizing every subspace in $S-\{V\}$ and not stabilizing $V$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19387 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Irredundant Generating Sets for Matrix Algebras Blumenthal, Yonatan First, Uriya Rings and Algebras Let $F$ be a field. We show that the largest irredundant generating sets for the algebra of $n\times n $ matrices over $F$ have $2n-1$ elements when $n>1$. (A result of Laffey states that the answer is $2n-2$ when $n>2$, but its proof contains an error.) We further give a classification of the largest irredundant generating sets when $n\in\{2,3\}$ and $F$ is algebraically closed. We use this description to compute the dimension of the variety of $(2n-1)$-tuples of $n\times n$ matrices which form an irredundant generating set when $n\in\{2,3\}$, and draw some consequences to Zariski-locally redundant generation of Azumaya algebras. In the course of proving the classification, we also determine the largest sets $S$ of subspaces of $F^3$ with the property that every $V\in S$ admits a matrix stabilizing every subspace in $S-\{V\}$ and not stabilizing $V$. |
| title | Irredundant Generating Sets for Matrix Algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2503.19387 |