Zinovy Reichstein and Nikolaus Vonessen, Group actions and invariants in algebras of generic matrices, Advances in Applied Mathematics 37, no.4 (2006), 481-500 (in the special issue in honor of Amitai Regev on his 65th birthday).

Abstract:

We show that the fixed elements for the natural GL_m-action on the universal division algebra UD(m, n) of m generic n x n-matrices form a division subalgebra of degree n, assuming n >= 3 and 2 <= m <= n² -2. This allows us to describe the asymptotic behavior of the dimension of the space of SL_m-invariant homogeneous central polynomials p(X_₁, ..., X__m) for  n x n-matrices. Here the base field is assumed to be of characteristic zero.