Dual pairs in PGL(n,ℂ)

2024/10/12 by Marisa Gaetz, Gaetz, Marisa
Computer Science · Engineering · Mathematics · #20G05 (Secondary) #22E10 (Primary) #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2410.09625

openalex publication_date 2024/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In Roger Howe's seminal 1989 paper "Remarks on classical invariant theory," he introduces the notion of Lie algebra dual pairs, and its natural analog in the groups context: a pair (G₁,G₂) of reductive subgroups of an algebraic group G is a dual pair in G if G₁ and G₂ equal each other's centralizers in G. While reductive dual pairs in the complex reductive Lie algebras have been classified, much less is known about algebraic group dual pairs, which were only fully classified in the context of certain classical matrix groups. In this paper, we classify the reductive dual pairs in PGL(n,ℂ).

Related

• Dual pairs in complex classical groups and Lie algebras
• An explicit classification of dual pairs in exceptional Lie algebras
• A classification of the principal nilpotent pairs in simple Lie algebras and related problems
• Reductive pairs arising from representations
• Orthogonal linear group-subgroup pairs with the same invariants