Flag varieties and interpretations of Young tableau algorithms

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.Comment: 16 page

Complementary Algorithms For Tableaux

We study four operations defined on pairs of tableaux. Algorithms for the first three involve the familiar procedures of jeu de taquin, row insertion, and column insertion. The fourth operation, hopscotch, is new, although specialised versions have appeared previously. Like the other three operations, this new operation may be computed with a set of local rules in a growth diagram, and it preserves Knuth equivalence class. Each of these four operations gives rise to an a priori distinct theory of dual equivalence. We show that these four theories coincide. The four operations are linked via the involutive tableau operations of complementation and conjugation.Comment: 29 pages, 52 .eps files for figures, JCTA, to appea

On the sign-imbalance of skew partition shapes

Let the sign of a skew standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. We examine how the sign property is transferred by the skew Robinson-Schensted correspondence invented by Sagan and Stanley. The result is a remarkably simple generalization of the ordinary non-skew formula. The sum of the signs of all standard tableaux on a given skew shape is the sign-imbalance of that shape. We generalize previous results on the sign-imbalance of ordinary partition shapes to skew ones.Comment: 14 pages; former section 8 is removed and the rest is slightly update

Enumeration of Standard Young Tableaux

A survey paper, to appear as a chapter in a forthcoming Handbook on Enumeration.Comment: 65 pages, small correction