A Fischer inequality for the second immanant

AbstractDenote by Hn the cone of n-by-n positive semidefinite Hermitian matrices. Let d2 be the generalized matrix function (or immanant) afforded by the symmetric group Sn and the irreducible degree n − 1 character corresponding to the partition (2,1,…,1). Suppose A ∈ Hn is partitioned into blocks, A=A11A12A∗12A22 If n⩾4, then d2(diag(A11,A22))⩾d2(A). This follows from the fact that d2 is a Schur-concave function of the spectrum of A for A ∈ Cn, where Cn consists of the matrices in Hn whose diagonal entries all equal 1