Summary: Let \(G\) be a reductive linear algebraic group over an algebraically closed field \(K\), let \(\widetilde P\) be a parabolic subgroup scheme of \(G\) containing a Borel subgroup \(B\), and let \(P=\widetilde P_{\text{red}}\subset\widetilde P\) be its reduced part. Then \(P\) is reduced, a variety, one of the well known classical parabolic subgroups. For \(\text{char} (K)=p<3\), a classification of the \(\widetilde P\)’s has been given by C. Wenzel in Trans. Am. Math. Soc. 337, No. 1, 211-218 (1993; Zbl 0785.20024).
The Chow ring of \(G/P\) only depends on the root system of \(G\). Corresponding to the natural projection from \(G/P\) to \(G/ \widetilde P\) there is a map of Chow rings from \(A(G/\widetilde P)\) to \(A(G/P)\), where \(A(G/P)\) denotes the Chow ring of \(G/P\). This map will be explicitly described here. Let \(P=B\), and let \(p>3\). A formula for the multiplication of elements in \(A(G/\widetilde P)\) will be derived. We will prove that \(A(G/ \widetilde P)\simeq A(G/P)\) (abstractly as rings) if and only if \(G/P\simeq G/ \widetilde P\) as varieties, i.e., the Chow ring is sensitive to the thickening. Furthermore, in certain cases \(A(G/ \widetilde P)\) is not any more generated by the elements corresponding to codimension one Schubert cells.