Richardson varieties in the Grassmannian

The Richardson variety X-w(v) is defined to be the intersection of the Schubert variety X-w and the opposite Schubert variety X-v. For X-w(v) in the Grassmannian, we obtain a standard monomial basis for the homogeneous coordinate ring of X-w(v). We use this basis first to prove the vanishing of H-i(X-w(v), L-m), i > 0, m greater than or equal to 0, where L is the restriction to X-w(v) of the ample generator of the Picard group of the Grassmannian; then to determine a basis for the tangent space and a criterion for smoothness for X-w(v) at any T-fixed point e(tau); and finally to derive a recursive formula for the multiplicity of X-w(v) at any T-fixed point e(tau). Using the recursive formula, we show that the multiplicity of X-w(v) at e(tau) is the product of the multiplicity of X-w at e(tau) and the multiplicity of X-v at e(tau). This result allows us to generalize the Rosenthal-Zelevinsky determinantal formula for multiplicities at T-fixed points of Schubert varieties to the case of Richardson varieties.