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In their study of local models of Shimura varieties for totally ramifiedextensions, Pappas and Rapoport posed a conjecture about the reducedness of acertain subscheme of $n \times n$ matrices. We give a positive answer to theirconjecture in full generality. Our main ideas follow naturally from two of ourprevious works. The first is our proof of a conjecture of Kreiman, Lakshmibai,Magyar, and Weyman on the equations defining type A affine Grassmannians. Thesecond is the work of the first two authors and Kamnitzer on affineGrassmannian slices and their reduced scheme structure. We also present aversion of our argument that is almost completely elementary: the onlynon-elementary ingredient is the Frobenius splitting of Schubert varieties.

On a conjecture of Pappas and Rapoport about the standard local model for $GL_d$