Norms on complex matrices induced by complete homogeneous symmetric polynomials

TL;DR

This paper introduces a novel family of matrix norms derived from symmetric functions, combining combinatorics, probability, and algebra to achieve unique properties and inequalities.

Contribution

It presents a new class of norms on complex matrices based on complete homogeneous symmetric polynomials, with applications to graph distinction and dimension-independent inequalities.

Findings

Norms have a determinantal interpretation.

They can distinguish certain graphs.

They lead to new tracial inequalities.

Abstract

We introduce a remarkable new family of norms on the space of $n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. Our norms enjoy many desirable analytic and algebraic properties, such as an elegant determinantal interpretation and the ability to distinguish certain graphs that other matrix norms cannot. Furthermore, they give rise to new dimension-independent tracial inequalities. Their potential merits further investigation.