In this article, we describe a way of removing the internal squares of a skew SSYT to turn it into an SSYT.
Definition. First write the skew diagram as ; we define an inside corner to be a square in such that there is no square to its right or below lying within
Note that there is some ambiguity in this definition, since there is generally more than one way of writing a skew diagram in the form For example if and then where and
In the left diagram, there is only one inside square at (2, 2) while in the right diagram there are two inside squares at (1, 2) and (2, 1). Henceforth, our construction is performed with respect to a fixed representation of the skew diagram as
Removing an Inside Corner
Now we describe the procedure. Let be a skew SSYT; pick any inside corner of a blank square. At each step, we look at its right and below.
- If there are numbers to its right and below, we swap with the smaller value; if both numbers are equal, the one below is taken to be smaller.
- If there is only one number to its right or below, we swap with that.
- If there are no numbers to its right and below, we can just remove .
Note that at each step, the rows are guaranteed to be weakly increasing and the columns strictly increasing, so at the end we get another skew tableaux but with one less empty square.
Lemma. The process is reversible, i.e. starting from the skew tableaux at the end, and an additional square at the end of a row such that the union is still a skew diagram, we can reverse the process of sliding.
One observes that at each step of the sliding process, the move applied is the only possible one such that the rows remains weakly increasing and columns strictly increasing, i.e. the move is the only legal one available.
The same holds for the reverse sliding process: at each step, we take the empty square and look at its left and above, then swap with the larger one; if both are equal, we swap with the one above. ♦
Definition. The sliding algorithm on a skew SSYT is as follows: we iteratively remove inner squares as above until we get an SSYT
It is a non-trivial result that the resulting is independent of our order of removal, or our choice of expression
There are two ways to perform the sliding algorithm on the following skew SSYT. Check that they both lead to the same SSYT.
Assuming the above uniqueness result, we define the following.
Definition. Given two SSYT and , their product is obtained by placing below and on the right to form a skew SSYT as follows:
then performing the sliding algorithm to turn it into an SSYT.
It is also true that the product is associative. All these will be proven within the next few articles.