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

TL;DR

This paper unifies and generalizes classical proofs of Schur's inequality by replacing contraction operators with projection operators, providing a clearer geometric perspective and extending the understanding of equality conditions.

Contribution

It introduces a unified framework using projection operators to analyze symmetric tensors and Schur's inequality, clarifying the geometric intuition behind the proofs.

Findings

Unified proof of Schur's inequality and equality conditions

Replaced contraction operators with projection operators for clarity

Extended understanding of symmetry and equality in tensor analysis

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.