Hiroshima Mathematical Journal

Sharp inequalities for the permanental dominance conjecture

Ryo Tabata

Full-text: Open access


For the normalized generalized matrix function $\overline d_{\chi}^{G} (A)$ for $3 \times 3$ positive semi-definite Hermitian matrices $A$, the permanental dominance conjecture $\per A \geq \overline d_{\chi}^{G} (A)$ is known to hold. In this paper, we show that this inequality is not sharp, and give a sharper bound.

Article information

Hiroshima Math. J. Volume 40, Number 2 (2010), 205-213.

First available in Project Euclid: 2 August 2010

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14] 20A30

symmetric group permanent generalized matrix function


Tabata, Ryo. Sharp inequalities for the permanental dominance conjecture. Hiroshima Math. J. 40 (2010), no. 2, 205--213. https://projecteuclid.org/euclid.hmj/1280754421.

Export citation


  • W. Barrett, H. Tracy Hall, R. Loewy, The cone of class function inequalities for the $4$-by-$4$ positive semidefinite matrices, Proc. London Math. Soc. (3) 79:107-130 (1999).
  • E. Fisher, Über den Hadamardschen Determinantensatz, Archiv d. Math. u. Phys. (3)13:32-40 (1907).
  • J. Hadamard, Resolution d'une question relative aux determinants, Bull. Sci. Math. 2:240-246 (1893).
  • P. Heyfron, Immanant dominance orderings for hook partitions, Linear and Multilinear Algebra 24(1):65-78 (1988).
  • P. Heyfron, Positive functions defined on Hermitian positive semi-definite matrices. Ph.D.thesis, University of London (1989).
  • G. D. James, Immanants, Linear and Multilinear Algebra 32:197-210 (1992).
  • G. D. James, Private communications.
  • E. H. Lieb, Proofs of some conjectures on permanents, I. Math. and Mech. 16:127-134 (1966).
  • M. Marcus, The Hadamard theorem for permanents, Proc. Amer. Math. Soc. 15:967-973 (1964).
  • T. H. Pate, Row appending maps, $\Psi$ functions, and immanant inequalities for Hermitian positive semi-definite matrices, Proc. London Math. Soc. (3) 76(2):307--358 (1998).
  • I. Schur, Über endliche Gruppen und Hermitische Formen, Math. Z. 1:184-207 (1918).
  • R. Tabata, A generalization of Schur's theorem. Master's thesis, Hiroshima University (2009).