Page 23 - Textos de Matemática Vol. 39

JEU DE TAQUIN, INVARIANT FACTORS 13
Lemma 3.1. (a) The transformation Σ ↔ Σ′, as above, establishes a bijective correspondence between the k-tuples of disjoint weakly increasing subwords of J = Jt ···J1 and those of decreasing subwords of ω = w1 ···wn.
(b) The SSYT’s P(ω) and P(J) have conjugate shapes with wtω the re- verse shape of the t-column word Jt ···J1, and wtJ the shape of the n-column word w1 ···wn.
We consider the variant of the dual RSK-correspondence, for short RSK∗, u RSK∗
defined and denoted by v −→ (P,Q), where P = P(w) and Q = P(J). The SSYT’s in this pair are related as follows, where v ↓ (u ↑) denotes v by
weakly decreasing (u by weakly increasing) order
Proposition 3.2. Let Σ = u↑ and Σ′ = J correspond by RSK∗
to (P,Q). Then ω v ↓
(a) Q is the unique SSYT of weight (|w1|, . . . , |wn|) such that
Q(ω) = std(Q)T , where std stands for standardization.
(b) P is the unique Q(J ) = std(evac P )T .
SSYT of evaluation (|J1 |, . . . , |Jt |) such that
The RSK∗ correspondence establishes a bijection between the biwords Σ (Σ′) over the alphabet [n] × [t] and tableau-pairs (P, Q) of conjugate shapes, with P ∈ [t]∗ and Q ∈ [n]∗. As we have seen in the previous section there is a bijection between column words and SSYT’s in the compact form. We have therefore the following bijections
1f1 ··· nfn {SSYT′s with n-column word ω = w1 ...wn} ↔ w1 ··· wn ↔
(3.1) ↔ Jt ··· J1 ↔ (P,Q) SSYT’s of conjugate shapes. tmt ··· 1m1
Identify a column word with its underlying set. The set Ji ⊆ [n] is the set of the column indices of the letter i in any SSYT with n-column word ω = w1 . . . wn. A SSYT defines a unique pair (ω, J), in the conditions of Propo- sition 3.2, and J = Jt . . . J1 is called the indexing set column word.
Let σ ∈ St and si the transposition of the integers i and i + 1. Put σ# := rev σ with rev the reverse permutation in St. In particular, s#i = st−i rev. If m is a weak composition, put σm for the usual action of σ on m. In particular, put m# := rev m. If J is a frank word, σJ denotes the frank word congruent with J whose shape is the action of σ on the shape of J. In particular, put J# := rev J. Given the sequence of nonnegative integers u = (u1,··· ,ur), we define the r-columnwordMu=u1 ···1u1+u2 ···u1+1··· ur+···+u1 ···ur−1+···+u1+1 [19] whose shape is u. The tableau-pair (K, Q) of conjugate shapes with K a