Jeu de taquin

In the mathematical field of combinatorics, jeu de taquin is a construction due to Marcel-Paul Schützenberger (1977) which defines an equivalence relation on the set of skew standard Young tableaux. A jeu de taquin slide is a transformation where the numbers in a tableau are moved around in a way similar to how the pieces in the fifteen puzzle move. Two tableaux are jeu de taquin equivalent if one can be transformed into the other via a sequence of such slides.

"Jeu de taquin" (literally "teasing game") is the French name for the fifteen puzzle.

Definition of a jeu de taquin slide

Example of a Jeu de taquin slide

Given a skew standard Young tableau T of skew shape $\lambda \setminus \mu$, pick an adjacent empty cell c that can be added to $\lambda \setminus \mu$; what this means is that c must share at least one edge with some cell in T, and $\{c\}\cup \lambda \setminus \mu$ must also be a skew shape. There are two kinds of slide, depending on whether c lies to the upper left of T or to the lower right. Suppose to begin with that c lies to the upper left. Slide the number from its neighbouring cell into c; if c has neighbours both to its right and below, then pick the smallest of these two numbers. (This rule is designed so that the tableau property of having increasing rows and columns will be preserved.) If the cell that just has been emptied has no neighbour to its right or below, then the slide is completed. Otherwise, slide a number into that cell according to the same rule as before, and continue in this way until the slide is completed. After this transformation, the resulting tableau is still a skew (or possibly straight) standard Young tableux.

The other kind of slide, when c lies to the lower right of T, just goes in the opposite direction. In this case, one slides numbers into an empty cell from the neighbour to its left or above, picking the larger number whenever there is a choice. The two types of slides are mutual inverses – a slide of one kind can be undone using a slide of the other kind.

The Schützenberger involution

Jeu de taquin can be used to define an operation on standard Young tableaux of any given given shape, which turns out to be an involution, although this is not obvious from the definition. One starts by emptying the square in the top-left corner, turning the tableau into a skew tableau with one less square. Now apply a jeu de taquin slide to turn that skew tableau into a straight one, which will free one square on the outside border. Then fill this square with the negative of the value that was originally removed at the top-left corner; this negated value is considered part of a new tableau rather than of the original tableau, and its position will not change in the sequel. Now as long as the original tableau has some entries left, repeat the operation of removing the entry x of the top-left corner, performing a jeu de taquin slide on what is left of the original tableau, and placing the value −x into the square so freed. When all entries of the original tableau have been handled, their negated values are arranged in such a way that rows and columns are increasing. Finally one can add an appropriate constant to all entries to obtain a Young tableau with positive entries.

Applications

Jeu de taquin is closely connected to such topics as the Robinson–Schensted–Knuth correspondence, the Littlewood–Richardson rule, and Knuth equivalence.

References

• Désarménien, J. (2001) [1994], "Jeu de taquin", Encyclopedia of Mathematics, EMS Press
• Sagan, Bruce E. (2001), The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Graduate Texts in Mathematics 203 (2nd ed.), New York: Springer, ISBN 0-387-95067-2
• Schützenberger, Marcel-Paul (1977), "La correspondance de Robinson", in Foata, Dominique (ed.), Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), Lecture Notes in Math., vol. 579, Berlin: Springer, pp. 59–113, doi:10.1007/BFb0090012, ISBN 978-3-540-08143-2 {{citation}}: More than one of |contribution= and |chapter= specified (help)

• Stanley, Richard P. (1999), Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics 62, vol. 2, Cambridge University Press, ISBN 0-521-56069-1