Schur functions and immanantal identities

Abstract

Littlewood developed the theory of symmetric functions and immanants. It is known that some identities for immanants correspond to the ordinary products of Schur functions via the Littlewood-Richardson rule. We discuss the relations between immanants and plethysm, another type of products of Schur functions. We present immanantal identities corresponding to the most basic formula of plethysm. As an application, we show some inequalities for positive semidefinite Hermitian matrices.