On Higman's k(U_n(\mathbbF_q)) conjecture

2015/07/02 by Igor Pak, Pak, Igor, Andrew Soffer +1 · 4 citations
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1507.00411

openalex publication_date 2015/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A classical conjecture by Graham Higman states that the number of conjugacy classes of U_n(q), the group of upper triangular n× n matrices over \mathbbF_q, is polynomial in q, for all n. In this paper we present both positive and negative evidence, verifying the conjecture for n≤ 16, and suggesting that it probably fails for n≥ 59. The tools are both theoretical and computational. We introduce a new framework for testing Higman's conjecture, which involves recurrence relations for the number of conjugacy classed of pattern groups. These relations are proved by the orbit method for finite nilpotent groups. Other applications are also discussed.

Cited by

• Supercharacter theories for Sylow p-subgroups of Lie type G₂
• The modality of a Borel subgroup in a simple algebraic group of type E₈
• Commuting matrices and volumes of linear stacks
• Rook placements and Jordan forms of upper-triangular nilpotent matrices

Related

• A Quiver Analogue of Higman's Conjecture
• Commuting matrices and volumes of linear stacks
• Upper bounds on the number of conjugacy classes in unitriangular groups
• The power conjugacy problem in Higman-Thompson groups
• Motivic Higman's Conjecture