Positive maps and trace polynomials from the symmetric group

With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve monomials Xα1,…,Xαr their traces tr(Xα1,…,Xαr). Our rests on translating action symmetric group tensor product spaces into multiplication. As result, extend polarized Cayley–Hamilton identity an inequality positive cone, characterize set multilinear equivariant maps terms Werner state witnesses, construct permutation polynomials polynomial spaces. We give connections concepts theory invariant theory.