Polynomials and Representations XIV

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 $\lambda/\mu$ ; we define an inside corner to be a square in $\mu$ such that there is no square to its right or below lying within $\mu.$

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 $\lambda/\mu.$ For example if $\lambda = (3, 2, 1)$ and $\mu = (2,2),$ then $\lambda/\mu = \lambda'/\mu'$ where $\lambda' = (3,1,1)$ and $\mu' = (2,1).$

equal_skew_tableaux

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 $\lambda/\mu.$

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Removing an Inside Corner

Now we describe the procedure. Let $T$ be a skew SSYT; pick any inside corner $S$ of $T,$ a blank square. At each step, we look at its right and below.

If there are numbers to its right and below, we swap $S$ 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 $S$ with that.
If there are no numbers to its right and below, we can just remove $S$ .

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.

Example

skew_tableaux_sliding

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.

Proof

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. ♦

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Sliding Algorithm

Definition. The sliding algorithm on a skew SSYT $T$ is as follows: we iteratively remove inner squares as above until we get an SSYT $T'.$

It is a non-trivial result that the resulting $T'$ is independent of our order of removal, or our choice of expression $\lambda / \mu.$

Exercise

There are two ways to perform the sliding algorithm on the following skew SSYT. Check that they both lead to the same SSYT.

skew_ssyt_exercise

Assuming the above uniqueness result, we define the following.

Definition. Given two SSYT $T$ and $T'$ , their product $T*T'$ is obtained by placing $T$ below and $T'$ 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.

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