Page 22 - Textos de Matemática Vol. 39

12 OLGA AZENHAS
The k-column word of a SSYT with k columns is defined as (u1, . . . , uk) ∈ V k with ui the i-th column. Two SSYT’s with k columns are said to be equiv- alent if they have the same k-column word. This means that they have the same compact form in the sense that every SSYT is jeu de taquin equivalent, by vertical slides, to its compact form. However, not every pair of SSYT’s with k columns, jeu de taquin equivalent, has the same compact form. For
6
instance (for convenience we represent the inner shape by X) 5 is in
X4
the compact form while X5 4 is not, but it is equivalent to the first one;
XX
6
and 5 is in the compact form but it is not equivalent to the previous
4X
ones, although jeu de taquin equivalent. The six-column word 53 52 76 ε 5 53 is
5
357
X 2 6 , with shape XXXX55 XXXXX3
identified with the six-column SSYT, (5,4,4,2,2,2)/(3,2,2,2,1,0).
6
3.2. A variant of the dual RSK-correspondence. We consider a vari- ant of the dual Robinson-Schensted-Knuth correspondence [13], [14], Appen- dix A.4.3, to establish a bijection between tableau-pairs (P,Q) of conjugate
shapesandSSYT’sinthecompactform.Let u = u1 ··· uk bea v v1···vk
biword without repeated biletters, where u1, . . . , uk ∈ [n] and v1, . . . , vk ∈ [t]. Sorting the biletters of uv by weakly increasing rearrangement for the anti-
1f1 ··· nfn lexicographic order with priority on the first row, we get Σ = w1 · · · wn ,
where wtu = (|w1|,...,|wn|) and ω = w1 ···wn ∈ Vn (here V is the set of columns of [t]∗); and by weakly decreasing rearrangement of the biletters of
uv for the lexicographic order with priority on the second row, we get Σ′= Jt ··· J1 ,wherewtv=(|J|,...,|J|)andJ=J···J ∈Vt
tmt···1m1 1t t1
(here V is the set of columns of [n]∗). As usual, given a word ω, P(ω) denotes the unique SSYT of partition shape in the Knuth class of ω, and Q(ω) the corresponding Q-symbol [22]. From Greene’s Theorem [11], we have