A functional interpretation of the ring of matrix invariants (Q750572)

Let k be an algebraically closed field of characteristic zero and \(C=C(n,m)\) be the ring of invariants of m-tuples of \(n\times n\) matrices over k. Some properties of the quotient space \(Q=Spec C\) are found in the paper. The main result (Theorem 3.3) states that if R is a commutative k- algebra and G(R) is the set of all irreducible representations of \(R\otimes L\) (L is the free associative algebra of rank m) in universal Cayley-Hamilton algebras then the functor G is represented by Q. Some sufficient conditions for a universal Cayley-Hamilton algebra having a unique trace function are given in the paper. A classification of universal Cayley-Hamilton algebras over an algebraically closed field is obtained.