Coincidentally, the most recent Richard III I’ve seen had him played by Seana McKenna and she was very, very good at being as bad as Billy quilled him to be.

]]>First, I got the impression from Bob’s talk that the third description using crystal graphs and Kashiwara operators, while the most complicated, was actually the most useful in constructing the rewriting system and automatic structure.

Second, I maybe didn’t make clear enough that elements of the plactic monoid correspond bijectively with semi-standard tableaux. So of course one should be able to define the monoid operation directly in terms of the tableaux. Indeed you can. To compose two tableaux, read the elements in the first tableau “Japanese style” as I described, and insert them in order into the second tableau (I think I have the order right, but because of using columns rather than rows I am not quite sure).

This also suggests an interesting problem. As I mentioned very briefly, the insertion algorithm gives a bijection between elements of the symmetric group and pairs of standard tableaux of the same shape. Can one define the group operation in the symmetric group directly in terms of pairs of tableaux? (This is quite different from the plactic monoid, since the group operation is substitution rather than concatenation.) Inversion is easy — just swap the two tableaux — but I have no idea how to do composition.

]]>I have also changed the wording of the post slightly at Jack’s suggestion.

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