Page 24 - Textos de Matemática Vol. 39

14 OLGA AZENHAS
key and the frank words in Q are related as follows. From Proposition 3.2 we have
Proposition 3.3. [9] Let (K,Q) be a tableau-pair of conjugate shapes such that the conjugate shape of Q is m. Let σ ∈ St. Let J correspond by
K↓ RSK∗ to the pair (K, Q). The following statements are equivalent
(a) K is the key with weight σm.
(b) J is the frank word of shape σ#m in the Knuth class of Q. (c) Q(J) = M(σ#m).
3.3. Jeu de taquin on two-column words. The jeu de taquin on consecu- tive columns t−i, t−i+1 of a t-column SSYT, in the compact form, exchanging the shape of these columns, is translated, by RSK∗-correspondence, into the operation θi on all words over the alphabet [t]. In particular, jeu de taquin on frank words is translated into the operation θi on words congruent with keys.
Define the operation Θ on a two-column SSYT in the compact form, with column lengths (q + s, q + r) and inner shape (r), for some q, r, s ≥ 0, as follows. If r > s (r < s), perform jeu de taquin slides on the first |r − s| inside (outside) corners until they become outside (inside) corners in the second (first) column. In other words, we slide down (up) the first (second) column, maximally up to |r − s| positions; then we exchange the east (west) neighbors with these corners. Then ΘT = T′ is a two-column SSYT, in the compact form, with column lengths (q + r, q + s) and with inner shape (s), if s ≥ r, and (r), otherwise. In particular, when r = 0 or s = 0, Θ is the jeu de taquin on frank words. For instance, the jeu de taquin slides with respect to the corner as below define the operation Θ on T and T ′
77 36 6
(3.2) T=25Θ 35. 1 4 ←→ T′= 2 4
3 13 22
Let T be a t-column SSYT, in the compact form. Define the operation Θi on T as follows: apply Θ to the columns i and i+1 of T and put the outcome t-column SSYT in the compact form. As jeu de taquin preserves Knuth equivalence, we have ΘiT ≡ T. The operations θi on words over the alphabet [t], and Θt−i on t-column semi-standard Young tableaux in the compact form (equivalence classes of SSYT’s) are a translation of each other in the sense of