2 - Words; the plactic monoid

Summary

In this chapter we study the word of a tableau, which encodes it by a sequence of integers. This is less visual than the tableau itself, but will be crucial to the proofs of the fundamental facts. Historically, however, the story goes the other way: the Schensted operations were invented to study sequences of integers. In this chapter we analyze what the bumping and sliding operations do to the associated words.

Words and elementary transformations

We write words as a sequence of letters (positive integers, with our conventions), and write w·w′ or ww′ for the word which is the juxtaposition of the two words w and w′.

Given a tableau or skew tableau T, we define the word (or row word) of T, denoted w(T) or w_row(T), by reading the entries of T “from left to right and bottom to top,” i.e., starting with the bottom row, writing down its entries from left to right, then listing the entries from left to right in the next to the bottom row and working up to the top. A tableau T can be recovered from its word: simply break the word wherever one entry is strictly greater than the next, and the pieces are the rows of T, read from bottom to top. For example, the word 5 6 4 4 6 2 3 5 5 1 2 2 3 breaks into 5 6 | 4 4 6 | 2 3 5 5 | 1 2 2 3, which is the word of a tableau used in examples in the preceding chapter.