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.