On Numbers of Tuples of Nilpotent Matrices over Finite Fields under Simultaneous Conjugation

2021/02/15 by Jiuzhao Hua, Hua, Jiuzhao
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Matrix Theory and Algorithms #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2102.07290

openalex publication_date 2021/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The problem of classifying tuples of nilpotent matrices over a field under simultaneous conjugation is considered "hopeless". However, for any given matrix order over a finite field, the number of concerned orbits is always finite. This paper gives a closed formula for the number of absolutely indecomposable orbits using the same methodology as Hua [5]; those orbits are non-splittable over field extensions. As a consequence, those numbers are always polynomials in the cardinality of the base field with integral coefficients. It is conjectured that those coefficients are always non-negative.

Citations

• Absolutely indecomposable representations and Kac-Moody Lie algebras (with an appendix by Hiraku Nakajima)

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