Flag Varieties and Interpretations of Young Tableau Algorithms

The conjugacy class of nilpotent n × n matrices can be parameterized by partitions λ of n, and for a nilpotent η in the class parameterized by λ, the variety η of η-stable flags has its irreducible components parameterized by the standard Young tableaux of shape λ. We indicate how several algorithmic constructions defined for Young tableaux have significance in this context, thus extending Steinbergʹs result that the relative position of flags generically chosen in the irreducible components of η 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 Schützenbergerʹs 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, 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 (as described in M. A. A. van Leeuwen, Electron. J. Combin.3, No. 2 (1996), R15).