2026/07/19 by Wei He, Wenhao Lu, Hang Yang +1
#math.NT
Given a finite dimensional representation π of a finitely generated group G=⟨ g1, …, gn⟩, the associated characteristic polynomial is defined as Qπ(z):=det(z0I+z1π(g1)+⋯ +znπ(gn)), and it is known to contain a good amount of structural information about G and π. This paper is a part of an ongoing project to investigate the number-theoretic properties of the algebraic varieties (called \em eigensurfaces) \z∈ ℂn+1: Qπ(z)=0\. Its focus is the distribution of prime triples in the eigensurface S:=\z∈ ℂ3: (z0+z1+z2)2+z0z1=0\ associated with the braid group \bf B3 and its reduced Burau representation. We prove that such triples occur with higher frequency on S than in the ambient lattice, revealing an unexpected connection between group representation theory and analytic number theory.