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 \(H^i(X_w^v,L^m)\), \(i > 0 \), \(m \geq 0\), 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 \(e_\t\); and finally to derive a recursive formula for the multiplicity of \(X_w^v\) at any \(T\)-fixed point \(e_\t\). Using the recursive formula, we show that the multiplicity of \(X_w^v\) at \(e_\t\) is the product of the multiplicity of \(X_w\) at \(e_\t\) and the multiplicity of \(X^v\) at \(e_\t\). 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.