Comparison of the JDT Emptying Procedure to the Evacuation and Promotion Procedures

In addition to the jeu de taquin emptying procedure, Schtzenberger defined two other procedures on skew tableaux in terms of the slideout operation. These are often called evacuation and promotion. Each of these can also be generalized to general posets. The three procedures inherit a similar flavor from the slideout operation,different papers have presented varying versions of them,and they have not been consistently named. Moreover,usage of the expression "jeu de taquin" has varied. The paragraphs below describe the most common usage of these terms. We do not know of any papers which consider the jeu de taquin emptying procedure in the setting of posets. Shapes and shifted shapes seem to be the only posets which have been noted to have the jeu de taquin property in published papers. There are only a few papers which use the slideout operation in the setting of posets, and each of those seems to be mainly concerned with the evacuation procedure.

We now describe the four procedures (slideout, jeu de taquin emptying, evacuation, promotion) in the context of shapes. Let M be a matrix of integers. Assume that the non-zero entries of M are distinct and strictly decrease within each row and within each column. If we choose a 0 located at (i,j), then moving this 0 consists of interchanging it with the larger of the two entries at (i+1,j) and (i,j+1). (If both of these entries are 0, then the operation is undefined.) To *slide out* a particular 0, we repeat the move operation for it for as long as this operation is defined. We call M a skew tableau if its 0's occur only in two regions: a northwestern region such that the entries above or to the left of any 0 are all 0's, and a southeastern region which is analogously convex to the east and to the south. It is a tableau if the northwestern region of 0's is empty.

Fix a skew tableau M and subscript all of the northwestern 0's with distinct positive integers which decrease across each row and column. The *jeu de taquin emptying procedure* consists of successively sliding the northwestern 0's out in the order of increasing subscripts and then disregarding all 0's. This produces a tableau T. The process is well defined in the sense that T is independent of the choice of subscripts.

Let T be a tableau. Replace the entry p at (1,1) with 0(p) and slide this 0 out. This entry is then left alone and this operation is repeated until every original entry q has been converted to 0(q) and slid out as far as possible. The result of *evacuation* is then obtained by replacing each 0(q) by -q. Remarkably, the evacuation process is an involution.

Now assume that all of the original entries of T are positive. Given such a T, replace the entry p at (1,1) with a 0 and slide this 0 out. Then replace this 0 at its final location with p- 2p. This negative entry will get shifted to the northwest as this operation is repeated for the other entries as they arrive at (1,1). This iteration continues until every original entry has been slid out and had 2p subtracted from it. The result of *promotion* is then obtained by adding 2p to every entry.