Boyle and Handelman have conjectured that whenever A is an n x n nonne
gative matrix with rank A less than or equal to r and Perron root lamb
da(1), the inequality det(lambda I - tA) less than or equal to lambda(
n-r)(lambda(r) - lambda(1)(r)) holds for all real numbers lambda satis
fying lambda greater than or equal to lambda(1). We introduce an analo
gous conjecture involving nonnegative central (class) functions on the
permutation group S-n. The analogue of the rank condition in this con
text is a condition on the support of the nonabelian Fourier transform
of the central function. We are able to establish that both conjectur
es are true in case 2r greater than or equal to n. (C) 1997 Elsevier S
cience Inc.