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arXiv · 2508.20199

The variety of nilpotent pairs $(A,B)$ with $[A,B] = \lambda I$

Vlad Roman·Robert M. Guralnick
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Abstract

Let $k$ be an algebraically closed field of characteristic $p >0$. We consider the variety of nilpotent pairs $(A,B)$ with $[A,B]=\lambda I$, namely the set of pairs $ X = \{ (A,B) \in M_n(k) \times M_n(k) \mid A,B \text{ nilpotent}, [A,B]=\lambda I, \lambda \in k \}$. We prove that if $n=pr$, then $X$ is irreducible of dimension $n^2$.

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Vlad Roman, Robert M. Guralnick. 2025-08-27. The variety of nilpotent pairs $(A,B)$ with $[A,B] = \lambda I$. https://arxiv.org/abs/2508.20199

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