Inequalities and identities for generalized matrix functions

Abstract

Let χ be a character on the symmetric group S n , and let A = ( a ij ) be an n-by- n matrix. The function d χ ( A) = Σ σϵS n χ( σ) Π n t = 1 a tσ( t) is called a generalized matrix function. If χ is an irreducible character, then d χ is called an immanent. For example, if χ is the alternating character, then d χ is the determinant, and if χ ≡ 1, then d χ is called the permanent (denoted per). Suppose that A is positive semidefinite Hermitian. We prove that the inequality (1/χ( id))d χ(A) ⪕ per A holds for a variety of characters χ including the irreducible ones corresponding to the partitions ( n − 1,1) and ( n − 2,1,1) of n. The main technique used to prove these inequalities is to express the immanents as sums of products of principal subpermanents. These expressions for the immanents come from analogous expressions for Schur polynomials by means of a correspondence of D.E. Littlewood.