We consider a desingularization Γ of a Richardson variety in the variety of complete flags, obtained as a fibre of a projection from a certain Bott-Samelson variety Z. For any projective embedding of Z via a complete linear system, we construct a basis of the homogeneous coordinate ring of Γ inside Z, indexed by combinatorial objects which we call w 0 -standard tableaux. and we have w max (i(J 6,1 )) = s 1 s 3 s 2 s 1 s 3 = [4231] = w 0 , hence T is not w 0 -standard.As an immediate consequence of Theorems 3.3 and 4.20, we have Corollary 4.22. Assume that m is regular, so we embed Γ i in the projective space P(H 0 (Γ i , L i,m ) * ). A basis of the homogeneous coordinate ring of Γ i is given by the w 0 -standard monomials of shape (i, dm) for all d ∈ N.

We show that in a cominuscule partial flag variety G/P , the multiplicity of an arbitrary point on a Richardson variety X v w = Xw ∩ X v ⊂ G/P is the product of its multiplicities on the Schubert varieties Xw and X v .