Page 25 - Textos de Matemática Vol. 39
P. 25
the diagram below Σ=T↑ .
Σ=
T
· · · i + 1q+siq+r · · ·
.
(3.3)
. . . . . . . . . . . . ... . . . .
. . . . . . . . . . . . ... . . . .
T′ ↑ θiw
′
Σ =
Θt−iT ···i+1q+riq+s···
Σ=
. . .
JEU DE TAQUIN, INVARIANT FACTORS
15
w . .
′
.
.
For instance, for the two-column SSYT T (3.2), we have the following diagram
Σ= 122334567 . 221211112 .
Σ =
T
23+1 13+2 .
Θ1T 23+213+1
,
1 2 2 3 3 4 5 6 7 221211122
′
Σ =
Σ =
. . . . . . . . . . . . . ... . . . .
. . . . . . . . . . . . . ... . . . .
.
.
. . .
′
where ω = (2(21)(21)1)122 → θ1ω = (2(21)(21)1)122.
As ΘiT ≡ T, from Proposition 3.2, the operator θi preserves the
Q-symbol, Q(ω) = Q(θiω). For a two-column SSYT T with r > s, we have, Q(T)=q+s+1 ··· 2q+s T
and
1 ··· q ··· q+s 2q+s+1 ··· 2q+s+r q+r+1 ··· 2q+r
Q(Θ1T)= 1 ··· q ··· q+r 2q+r+1 ··· 2q+s+r From Proposition 3.2,
T
.
and
Thus P(θiω) = θiP(ω).
Q(T ) = std(evac P (ω))T
Q(Θt−iT ) = std(evac P (θiω))T .
