Some remarks on a conjecture of Boyle and Handelman. (English) Zbl 0912.15019
Summary: M. Boyle and D. Handelman [Ann. Math., II. Ser. 133, No. 2, 249-316 (1991; Zbl 0735.15005)] have conjectured that whenever \(A\) is an \(n\times n\) nonnegative matrix with rank \(A\leq r\) and Perron root \(\lambda_1\), the inequality \(\text{det} (\lambda {\mathbf I}- tA)\leq \lambda^{n-r} (\lambda^r -\lambda^r_1)\) holds for all real numbers \(\lambda\) satisfying \(\lambda\geq \lambda_1\). We introduce an analogous conjecture involving nonnegative central (class) functions on the permutation group \(S_n\). 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 \(2r\geq n\).
MSC:
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
Keywords:
inequalities involving eigenvalues; nonnegative matrix; Perron root; rank condition; Fourier transformCitations:
Zbl 0735.15005References:
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