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THE CONE OF CLASS FUNCTION INEQUALITIES FOR THE 4-BY-4 POSITIVE SEMIDEFINITE MATRICES

Published online by Cambridge University Press:  01 July 1999

WAYNE BARRETT ,
H. TRACEY HALL  and
RAPHAEL LOEWY
WAYNE BARRETT
Affiliation:
Department of Mathematics , Brigham Young University, Provo, UT 84602, U.S.A. E-mail: [email protected]
H. TRACEY HALL
Affiliation:
Department of Mathematics , Brigham Young University, Provo, UT 84602, U.S.A. E-mail: [email protected]
RAPHAEL LOEWY
Affiliation:
Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel E-mail: [email protected] [email protected]
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Abstract

A function $f$ from the symmetric group $S_n$ into $\Bbb R$ is called a class function if $f(\sigma ) = f(\tau )$ whenever $\tau$ is conjugate to $\sigma$. Let $d_f$ be thegeneralized matrix function associated with $f$, mapping the $n$-by-$n$ positive semidefinite Hermitian matricesto $\Bbb R$. For example, if $f(\sigma ) = \mbox{sgn}(\sigma )$, then $d_f(A) = \det A$.

We consider the cone $K_n$ of those $f$ for which $d_f(A) \geq 0$ for all $n$-by-$n$ positive semidefinite Hermitian matrices. For $n = 4$ we show that $K_n$ is polyhedral, and explicitly find the extreme rays. Equivalently, $f$ belongs to $K_n$ if $d_f(A) \geq 0$ for a finite minimal set of ‘test matrices’. This solves a problem posed by Gordon James. As a corollary, we characterize all inequalities involving linear combinations of immanants on the positive semidefinite Hermitian matrices of order $n = 4$.

We obtain similar results for 4-by-4 real positive semidefinite matrices.

Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 79 , Issue 1 , July 1999 , pp. 107 - 130
Copyright
London Mathematical Society 1999

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