Recursive projections of symmetric tensors and Marcus's proof of the Schur inequality

Abstract

In a 1918 paper Schur proved a remarkable inequality that related group representations, Hermitian forms and determinants. He also gave concise necessary and sufficient conditions for equality. In 1964, Marcus gave a beautiful short proof of Schur's inequality by applying the Cauchy-Schwarz inequality to symmetric tensors, but he did not discuss the case of equality. In 1969, Williamson gave an inductive proof of Schur's equality conditions by contracting Marcus's symmetric tensors onto lower dimensional subspaces where they remained symmetric tensors. Here we unify these results notationally and conceptually, replacing contraction operators with the more geometrically intuitive projection operators.