Standard monomial theory and toric degenerations of Richardson varieties inside Grassmannians and flag varieties

Abstract

We study toric degenerations of opposite Schubert and Richardson varieties inside degenerations of Grassmannians and flag varieties. These degenerations are parametrized by matching fields in the sense of Sturmfels and Zelevinsky. We construct so-called restricted matching field ideals whose generating sets are understood combinatorially through tableaux. We determine when these ideals are toric and coincide with Gr\"obner degenerations of Richardson varieties using the well established standard monomial theory for Grassmannians and flag varieties.