A Fischer inequality for the second immanant
type: Item
instance of: scholarly article
title: A Fischer inequality for the second immanant
zbMATH Open document ID: 0608.15003
DOI: 10.1016/0024-3795(87)90159-5
author: Russell Merris
author: Robert D. Grone
published in: Linear Algebra and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Denote by \({\mathcal H}_ n\) the cone of n-by-n positive semidefinite Hermitian matrices. Let \(d_ 2\) be the generalized matrix function (or immanant) afforded by the symmetric group \(S_ n\) and the irreducible degree n-1 character corresponding to the partition (2,1,...,1). Suppose \(A\in {\mathcal H}_ n\) is partitioned into blocks. \[ A=\left[ \begin{matrix} A_{11}\\ A^*_{12}\end{matrix} \begin{matrix} A_{12}\\ A_{22}\end{matrix} \right]. \] If \(n\geq 4\), then \(d_ 2(diag(A_{11},A_{22}))\geq d_ 2(A)\). This follows from the fact that \(d_ 2\) is a Schur-concave function of the spectrum of A for \(A\in {\mathcal C}_ n\), where \({\mathcal C}_ n\) consists of the matrices in \({\mathcal H}_ n\) whose diagonal entries all equal 1.
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 3983385
zbMATH Keywords: Fischer inequality
zbMATH Keywords: positive semidefinite Hermitian matrices
zbMATH Keywords: generalized matrix function
zbMATH Keywords: immanant
zbMATH Keywords: irreducible degree
zbMATH Keywords: Schur-concave function
MaRDI profile type: Publication
full work available at URL: 0024-3795(87)90159-5
OpenAlex ID: W1991047965
cites work: Q5831338
cites work: An inequality for the second immanant
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cites work: Products and Plethysms of Characters with Orthogonal, Symplectic and Symmetric Groups
cites work: Q5331504
cites work: Inequalities: theory of majorization and its applications
cites work: Inequalities and identities for generalized matrix functions
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description: scientific article; zbMATH DE number 3983385