Group actions and invariants in algebras of generic matrices

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 × n -matrices form a division subalgebra of degree n, assuming n ⩾ 3 and 2 ⩽ m ⩽ n 2 − 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 × n -matrices. Here the base field is assumed to be of characteristic zero.