Sharp inequalities for the permanental dominance conjecture
Ryo Tabata
Source: Hiroshima Math. J. Volume 40, Number 2
(2010), 205-213.
Abstract
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.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.hmj/1280754421
Mathematical Reviews number (MathSciNet): MR2680656
Zentralblatt MATH identifier: 05831949
References
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).
Mathematical Reviews (MathSciNet): MR1687543
Zentralblatt MATH: 1024.15008
Digital Object Identifier: doi:10.1112/S0024611599011909
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).
Mathematical Reviews (MathSciNet): MR1007246
Zentralblatt MATH: 0678.15009
Digital Object Identifier: doi:10.1080/03081088808817899
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).
Mathematical Reviews (MathSciNet): MR1238004
Zentralblatt MATH: 0759.15003
Digital Object Identifier: doi:10.1080/03081089208818163
G. D. James, Private communications.
E. H. Lieb, Proofs of some conjectures on permanents, I. Math. and Mech. 16:127-134 (1966).
Mathematical Reviews (MathSciNet): MR202745
Zentralblatt MATH: 0144.26802
M. Marcus, The Hadamard theorem for permanents, Proc. Amer. Math. Soc. 15:967-973 (1964).
Mathematical Reviews (MathSciNet): MR168585
Zentralblatt MATH: 0166.29903
Digital Object Identifier: doi:10.1090/S0002-9939-1964-0168585-9
JSTOR: links.jstor.org
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).
Mathematical Reviews (MathSciNet): MR1490240
Zentralblatt MATH: 0907.15011
Digital Object Identifier: doi:10.1112/S0024611598000100
I. Schur, Über endliche Gruppen und Hermitische Formen, Math. Z. 1:184-207 (1918).
Mathematical Reviews (MathSciNet): MR1544291
Digital Object Identifier: doi:10.1007/BF01203611
R. Tabata, A generalization of Schur's theorem. Master's thesis, Hiroshima University (2009).
Hiroshima Mathematical Journal
