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TitleLocal properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence.
Author(s) Victor Kreiman
TypeArticle in Journal
AbstractThe Richardson variety X α γ in the Grassmannian is defined to be the intersection of the Schubert variety X γ and opposite Schubert variety X α . We give an explicit Gröbner basis for the ideal of the tangent cone at any T-fixed point of X α γ , thus generalizing a result of Kodiyalam-Raghavan (J. Algebra 270(1):2854, 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 α γ 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. 22:273288, 2005).
KeywordsSchubert variety, Grassmannian, Multiplicity
ISSN0925-9899; 1572-9192/e
URL http://link.springer.com/article/10.1007%2Fs10801-007-0093-0
JournalJ. Algebr. Comb.
PublisherSpringer US, New York, NY
Translation No
Refereed No