Geodesic
Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence
Journal of Algebraic Combinatorics, Tome 27 (2008) no. 3, pp. 351-382.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: The Richardson variety $X _{ \alpha } ^{ \gamma }$ in the Grassmannian is defined to be the intersection of the Schubert variety $X ^{ \gamma }$ and opposite Schubert variety $X _{ \alpha }$. We give an explicit Gröbner basis for the ideal of the tangent cone at any $T$-fixed point of $X _{ \alpha } ^{ \gamma }$, 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 $X _{ \alpha } ^{ \gamma }$ 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).
Keywords: keywords Schubert variety, Grassmannian, multiplicity, Gröbner basis, Robinson-Schensted-knuth correspondence 
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     author = {Kreiman, Victor},
     title = {Local properties of {Richardson} varieties in the {Grassmannian} via a bounded {Robinson-Schensted-Knuth} correspondence},
     journal = {Journal of Algebraic Combinatorics},
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Kreiman, Victor. Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence. Journal of Algebraic Combinatorics, Tome 27 (2008) no. 3, pp. 351-382. http://geodesic.mathdoc.fr/item/JAC_2008__27_3_a1/