September 24, 20250 citationsOpen Access

The variety of nilpotent pairs (A, B) with A, B = λI

Key Points

The variety of nilpotent pairs $(A,B)$ with $[A,B]=λI$ is shown to be irreducible.
For nilpotent matrices of size $n=pr$, the dimension of the variety is proven to be $n^2$.
The research focuses on pairs of matrices in the context of algebraically closed fields.
The study enhances the understanding of commutator relations within nilpotent algebra.

Abstract

Let k be an algebraically closed field of characteristic p >0. We consider the variety of nilpotent pairs (A, B) with A, B=λI, namely the set of pairs X = \ (A, B) Mₙ (k) Mₙ (k) A, B nilpotent, A, B=λI, λ k \. We prove that if n=pr, then X is irreducible of dimension n².

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Roman et al. (2025) studied this question.

synapsesocial.com/papers/68d6d82e8b2b6861e4c3e12d https://doi.org/10.48550/arxiv.2508.20199

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September 24, 20250 citationsOpen Access

The variety of nilpotent pairs (A, B) with A, B = λI

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VRVlad Roman RGRobert M. Guralnick

Key Points

The variety of nilpotent pairs $(A,B)$ with $[A,B]=λI$ is shown to be irreducible.
For nilpotent matrices of size $n=pr$, the dimension of the variety is proven to be $n^2$.
The research focuses on pairs of matrices in the context of algebraically closed fields.
The study enhances the understanding of commutator relations within nilpotent algebra.

Abstract

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Cite This Study

Roman et al. (2025) studied this question.

synapsesocial.com/papers/68d6d82e8b2b6861e4c3e12d https://doi.org/10.48550/arxiv.2508.20199

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