** Abstract: **
We show that the fixed elements for the natural
GLm-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 <= n^2 -2. This allows us to give
an asymptotic estimate on the dimension of the space of
SLm-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.