## Mathematics, Department of

## Date of this Version

7-5-2023

## Citation

*Forum of Mathematics, Sigma* (2023), Vol. 11:e67 1β43 doi:10.1017/fms.2023.67

## Abstract

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 inWeylβs 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 π^{πΊ} β π being the natural embedding.

Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring π^{πΊ} is a pure subring of *S*, equivalently, π^{πΊ} is a direct summand of *S* as an π^{πΊ}-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 π^{πΊ} β π is pure. It turns out that if π^{πΊ} β π is pure, then either the invariant ring π^{πΊ} is regular or the group *G* is linearly reductive.

## Comments

Open access.