Page 28 - Textos de Matemática Vol. 39

18 OLGA AZENHAS
Since ∆ = paqI, it suffices to analyze the following situation. Let aq
m = |J1 ∩ J2|, we have
DD∼DD(I⊕D )↔F=2...2
J1 J2 [|J1|] [m] |J1| |J2|−m 1...1 1|J1|−m 2|J2|−m and
DD∼DD(I⊕D )↔tF=2...2
J2 J1 [|J2|] [m] |J2| |J1|−m 1 1...1 1|J2|−m 2|J1|−m
= evacF,
where ∼ denotes unimodular equivalence.
Recall the operation Θ on a two-column semistandard Young tableau T in
the compact form, defined in the previous section, and denote by Θ˜ , a variant of Θ, based on a variant of the jeu de taquin on T, running as follows: (1) If r−s = r (s−r = s), add q extra vacant positions to the top (bottom) of the first (second) column and slide the labeled entries in the first (second) column along the enlarged column such that the row weak increasing order is preserved and a common label to the two columns never has a vacant west (east) neighbor; then mark r vacant positions in the first (second) column with labeled east (west) neighbors and exchange with them. When the labeled entries of the first (second) column are slided down (up) maximally such that the row weakly order is preserved, we get the jeu de taquin operation Θ on frank words. (2) If 0 ≤ r − s < r (0 ≤ s − r < s), add q vacant positions to the top (bottom) of the first (second) column and slide the q + s labeled entries along the enlarged first column with q + r vacant positions such that the previous restrictions are attained; then mark r−s vacant positions in the first (second) column with east (west) labeled neighbors and exchange with them. When the labeled entries of the first (second) column are slided down (up) maximally at most r−s positions such that the row weakly order is preserved, we get the operation jeu de taquin Θ o n T . U n l e s s , Θ˜ = Θ , Θ˜ T ̸ ≡ T . F o r i n s t a n c e ,
(4.1)
( 4 . 2 )
55
4 Θ 4 .
1 32 ←→ 1 23 55555
Θ˜ : 1 4 − → 2 4 Θ˜ : 1 4 − → 1 4 − → 3 4 . 313 3312
222