Flag varieties and interpretations of Young tableau algorithms

Abstract

The conjugacy classes of nilpotent $n\times n$ matrices can be parametrised by partitions $\lambda$ of $n$, and for a nilpotent $\eta$ in the class parametrised by $\lambda$, the variety $F_\eta$ of $\eta$-stable flags has its irreducible components parametrised 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 flags generically chosen in the irreducible components of $F_\eta$ parametrised 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 Sch\"utzenberger'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 flag (or pair of flags) determined by a point chosen in the variety under consideration. We show that for generic choices, the family satisfies certain combinatorial relations, whence the family describes the computation of the algorithmic operation being interpreted, as we described in a previous publication.