Standard monomial theory and toric degenerations of Richardson varieties in the Grassmannian

Richardson varieties are obtained as intersections of Schubert and opposite varieties. We provide a new family toric degenerations inside Grassmannians by studying Gröbner their corresponding ideals. These parametrised block diagonal matching fields in the sense Sturmfels [33]. associate weight vector to each field study its initial ideal. In particular, we characterise when such ideals toric, hence providing for Given variety $$X_{w}^v$$ $$\mathbf{w}_\ell $$ arising from field, consider two ideals: an ideal $$G_{k,n,\ell }|_w^v$$ restricting Plücker smaller polynomial ring, defined kernel monomial map $$\phi _\ell |_w^v$$ . first monomial-free form Then construct tableaux bijection with semi–standard Young which leads basis quotient ring. Finally, prove that is $$_{\mathbf{w}_\ell }(I(X_w^v))$$ quadratically generated, then all three , ker $$(\phi |_w^v)$$ coincide, degeneration $$X_w^v$$