Generalized matrix valued invariants (Q1320258)

Let \(X_{mn}\) be the space of \(m\)-tuples of \(n \times n\) matrices over \(\mathbb{C}\). The group \(\text{PSL}_ n (\mathbb{C})\) acts on \(X_{mn}\) by componentwise conjugation and this induces an action on the symmetric algebra \(S\) of \(X_{mn}\), i.e. on the polynomial algebra in \(mn^ 2\) variables. The action of \(\text{PSL}_ n(\mathbb{C})\) on \(S\) can be extended to \(M_ n(S)\) and to the \(r\)-th tensor power \(M_ n (S)^{\otimes r}\) in such a way that the generic \(n \times n\) matrices are invariants. It turns out that the algebra of the \(\text{PSL}_ n (\mathbb{C})\)-invariants of \(M_ n(S)\) is the generic trace ring \(T_{mn}\). The purpose of the paper under review is to prove a structure result on the invariants of \(M_ n(S)^{\otimes r}\) and shows how this invariant algebra relates to \((T^{\otimes r}_{mn})^{**}\). As consequences, the author generalizes the Artin-Schofield theorem on the height one prime ideals of \(S\) and gives a description of the projective locus of \(T_{mn}\).