Some remarks on a conjecture of Boyle and Handelman

Boyle and Handelman have conjectured that whenever A is an n × n nonnegative matrix with rank A r and Perron root λ1, the inequality det(λI − tA) λn−r(λr−λ1r) holds for all real numbers λ satisfying λ λ1. We introduce an analogous conjecture involving nonnegative central (class) functions on the permutation group Sn. The analogue of the rank condition in this context is a condition on the support of the nonabelian Fourier transform of the central function. We are able to establish that both conjectures are true in case 2r n.