Flag varieties and interpretations of Young tableau algorithms

JOURNAL OF ALGEBRA

卷:224

Article

van Leeuwen, MAA

关键词: flag manifold;    nilpotent;    Jordan decomposition;    jeu de taquin;    Robinson-Schensted correspondence;    Littlewood-Richardson rule;   

DOI  :  10.1006/jabr.1999.8070

来源: Elsevier

【 摘 要 】

The conjugacy class of nilpotent nxn matrices can be parameterized by partitions lambda of n, and for a nilpotent eta in the class parameterized by lambda, the variety F-n Of eta-stable flags has its irreducible components parameterized by the standard Young tableaux of shape lambda. We indicate how several algorithmic constructions defined for Young tableaux have significance in this context, thus extending Steinberg's result that the relative position of nags generically chosen in the irreducible components of F-eta parameterized by tableaux P and Q is the permutation associated to (P,Q) under the Robinson-Schensted correspondence. Other constructions for which we give interpretations are Schutzenberger's involution of the set of Young tableaux, jeu de taquin (leading also to an interpretation of Littlewood-Richardson coefficients), and the transpose Robinson-Schensted correspondence (defined using column insertion). In each case we use a doubly indexed family of partitions, defined in terms of the nag (or pair of flags) determined by a point chosen in the variety under consideration, and we show that for generic choices, the family satisfies combinatorial relations that make it correspond to an instance of the algorithmic operation being interpreted las described in M. A. A. van Leeuwen, Electron. J. Combin. 3, No. 2 (1996), R15). (C) 2000 Academic Press.

【 授权许可】

Free