When are the natural embeddings of classical invariant rings pure?

Abstract Consider a reductive linear algebraic group G acting linearly on polynomial ring S over an infinite field; key examples are the general group, symplectic orthogonal and special with classical representations as in Weyl’s book: For consider direct sum of copies standard representation dual; other cases, take representation. The invariant rings respective cases determinantal rings, defined by Pfaffians alternating matrices, symmetric Plücker coordinate Grassmannians; these title, $S^G\subseteq S$ being natural embedding. Over field characteristic zero, is reductive, it follows that $S^G$ pure subring , equivalently, summand -module. fields positive characteristic, groups typically no longer reductive. We determine, case, precisely when inclusion pure. It turns out if pure, then either regular or