Consider a reductive linear algebraic group $G$ acting linearly on a polynomial ring $S$ over an infinite field. Objects of broad interest in commutative algebra, representation theory, and algebraic geometry like generic determinantal rings, Plücker coordinate rings of Grassmannians, symmetric determinantal rings, rings defined by Pfaffians of alternating matrices etc. arise as the invariant rings $S^G$ of such group actions.

In characteristic zero, reductive groups are linearly reductive and therefore the embedding of the invariant ring $S^G$ in the ambient polynomial ring $S$ splits. This explains a number of good algebro-geometric properties of the invariant ring in characteristic zero. In positive characteristic, reductive groups are typically no longer linearly reductive. We determine, for the natural actions of the classical groups, precisely when $S^G$ splits from $S$ in positive characteristic.

This is joint work with Melvin Hochster, Jack Jeffries, and Anurag K. Singh.