Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence

The Richardson variety Xαγ in the Grassmannian is defined to be the intersection of the Schubert variety Xγ and opposite Schubert variety Xα. We give an explicit Gröbner basis for the ideal of the tangent cone at any T-fixed point of Xαγ, thus generalizing a result of Kodiyalam-Raghavan (J. Algebra 270(1):28–54, 2003) and Kreiman-Lakshmibai (Algebra, Arithmetic and Geometry with Applications, 2004). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the bounded RSK (BRSK). We use the Gröbner basis result to deduce a formula which computes the multiplicity of Xαγ at any T-fixed point by counting families of nonintersecting lattice paths, thus generalizing a result first proved by Krattenthaler (Sém. Lothar. Comb. 45:B45c, 2000/2001; J. Algebr. Comb. 22:273–288, 2005).