Edge Sequences, Ribbon Tableaux, and an Action of Affine Permutations

AbstractAn overview is provided of some of the basic facts concerning rim hook lattices and ribbon tableaux, using a representation of partitions by their edge sequences. An action is defined for the affine Coxeter group of type Ãr−1on ther-rim hook lattice, and thereby on the sets of standard and semistandardr-ribbon tableaux, and this action is related to the concept of chains inr-ribbon tableaux

Some bijective correspondence involving domino tableaux

We elaborate on the results in ``Splitting the square of a Schur function into its symmetric and antisymmetric parts '' [Carre Leclerc, J. algebr. combinat. 4, 1995]. We give bijective proof of a number of identities that were established there, in particular between the Yamanouchi domino tableaux, and the ordinary Littlewood-Richardson fillings that correspond to the same tensor product decomposition

The Littlewood-Richardson rule, theory and implementation

We present the implementation of the Littlewood-Richardson rule in LiE. We describe the mathematical problem it applies to, formulate the rule, and indicate a proof. In a brief historical sketch we indicate some early formulations and partial proofs. We derive a formulation of the rule that can be implemented very efficiently

Edge sequences, ribbon tableaux, and an action of affine permutations

An overview is provided of some of the basic facts concerning rim hook lattices and ribbon tableaux, using a representation of partitions by their edge sequences. An action is defined of the affine Coxeter group of type tilde A_{r-1 on the $r$-rim hook lattice, and thereby on the sets of standard and semistandard ribbon tableaux, and this action is related to the concept of chains in $r$-ribbon tableaux

Tableau algorithms defined naturally for pictures

We consider pictures as defined by Zelevinsky. We elaborate on the generalisation of the Robinson-Schensted correspondence to pictures defined by him, and on the result of Fomin and Greene that shows that this correspondence is natural, i.e., independent of the precise ``reading'' order of the squares of skew diagrams that is used in its definition. We give a simplified proof of this result by showing that the generalised Schensted insertion procedure can be defined without using this order at all. Our main results involve the operation of glissement defined by Schützenberger. We show that glissement can be generalised to pictures, and is natural. In fact, we obtain two dual forms of glissement; consequently both tableaux corresponding to a permutation in the Robinson-Schensted correspondence can be obtained by glissement from one picture. We show that the two forms of glissement commute with each other. From this fact the main properties of glissement follow in a much simpler way than their original derivation by Schützenberger