SOME REMARKS ON A CONJECTURE OF BOYLE AND HANDELMAN

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.