Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T06:42:11.065Z Has data issue: false hasContentIssue false
  • English
  • Français

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: loewy@techunix.technion.ac.il
H. TRACEY HALL
Affiliation:
Department of Mathematics , Brigham Young University, Provo, UT 84602, U.S.A. E-mail: loewy@techunix.technion.ac.il
RAPHAEL LOEWY
Affiliation:
Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel E-mail: wayne@math.byu.edu hthall@math.berkeley.edu
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)