Mathematics, Department of


Date of this Version



Forum of Mathematics, Sigma (2023), Vol. 11:e67 1–43 doi:10.1017/fms.2023.67


Open access.


Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as inWeyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the PlΓΌcker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with 𝑆𝐺 βŠ† 𝑆 being the natural embedding.

Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring 𝑆𝐺 is a pure subring of S, equivalently, 𝑆𝐺 is a direct summand of S as an 𝑆𝐺-module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion 𝑆𝐺 βŠ† 𝑆 is pure. It turns out that if 𝑆𝐺 βŠ† 𝑆 is pure, then either the invariant ring 𝑆𝐺 is regular or the group G is linearly reductive.