Consider the general linear group acting on a polynomial ring by the direct sum of copies of the standard representation and copies of its dual. The invariant ring of this action is the generic determinantal ring. When the underlying field has characteristic zero, this inclusion of the invariant ring in the ambient polynomial ring splits since the general linear group is linearly reductive. This has some rather strong consequences: the generic determinantal ring is a Cohen--Macaulay normal domain with rational singularities.
Does this inclusion also split when the underlying field is an infinite field of positive characteristic? Notably, the general linear group is typically not linearly reductive in positive characteristic.
Starting from Hilbert's foundational work in invariant theory, we will answer this question for all families of classical groups and discuss some consequences.
