September 24, 20250 citationsOpen Access

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

VRVlad RomanUniversity of Southern California RGRobert M. GuralnickUniversity of Southern California

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.

Ask AI

Helpful

Bookmark

View Full Paper

Cite This Study

Roman et al. (2025) studied this question.

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

Ask AI

Helpful

Bookmark

View Full Paper

September 24, 20250 citationsOpen Access

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

View Full Paper

VRVlad RomanUniversity of Southern California RGRobert M. GuralnickUniversity of Southern California

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

Ask AI

Helpful

Bookmark

View Full Paper

Cite This Study

Roman et al. (2025) studied this question.

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

Ask AI

Helpful

Bookmark

View Full Paper