Page 26 - Textos de Matemática Vol. 39

16 OLGA AZENHAS
Theorem 3.4. Given Q a SSYT, let J be a t-column word congruent with Q.LetP beaSSYTofweightthereverseshapeofJ.IfJ andQ↑
P↓ w are biwords in RSK∗ correspondence with (P, Q), then Θt−iJ and Q ↑
correspond by RSK∗ to (θiP, Q). θiP ↓ θiw
Corollary 3.5. [9] Given Q a SSYT and σ ∈ St, let K be the key tableau of
weight the shape of σQ. If σ#Q and Q ↑ are biwords in RSK∗ corre- K↓ω
spondence with (K, Q), then st−iσ#Q and Q ↑ correspond by RSK∗ to
(θiK,Q). θiK ↓ θiω
Remark3.6. IfJ=J2J1 isatwo-columnfrankwordwithJ2 =y+q+r+ s...y+q+r+1...y+q+1...q+1 and J1 = y+q+r+1...y+q+1...y+1, we have ω = P (ω) a key tableau and θ1P (ω) = t1P (ω) (see also [5]).
Corollary 3.7. The following statements are equivalent:
(a) The operations Θi, 1 ≤ i ≤ t − 1, define an action of the symmetric
group St on the set of t-column words, equivalently, on the t-column SSYT’s in the compact form. Moreover, ΘiT ≡ T , 1 ≤ i ≤ t − 1.
(b) The operations θi, 1 ≤ i ≤ t − 1, defines an action of the symmetric group on all words over the alphabet [t]. These operations preserve the Q-symbol andω≡P iffθiω≡θiP.
Example 3.8. An action of S3 on three-column SSYT’s in the compact form 99
777 266 267 145 146
9 7
2 3 . 1 2
.
2 4 1 3 2
9 .. 7
Θ1 6 .
. .
.
. Θ1
. . .
. . .
1 4 3
2 Θ2 .
7
6 7
7 7 6 6
1 4 3 2
..
. 6
34Θ2 35
.
47
236 1259925
.. .
. . .
. .
. .
. .
. .
. .
. .
.
. .
. . .
Θ
.. . .
.. . .
2
46 . Θ1 . 245 .
. . .
235 134 124 23 13 12
2
247 136.