A Fischer inequality for the second immanant
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.
positive semidefinite Hermitian matrices
scientific article; zbMATH DE number 3983385
scientific article