Group actions and invariants in algebras of generic matrices
Zinovy Reichstein, Nikolaus Vonessen
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 <= 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 x n matrices. Here the base field is assumed to be of characteristic zero.
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