Norms on complex matrices induced by complete homogeneous symmetric polynomials

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