Group actions and invariants in algebras of generic matrices

We show that the fixed elements for the natural GLm-action on the universal division algebra UD(m,n) of m generic n×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 SLm-invariant homogeneous central polynomials p(X1,…,Xm) for n×n-matrices. Here the base field is assumed to be of characteristic zero.