Page 32 - Textos de Matemática Vol. 39

22 OLGA AZENHAS
Theorem 4.9. Let ∆μ,UDF be a matrix realization of T with word ω and indexing set word J2 J1 , and let ∆μ , U Dθ1 F be a matrix realization of T ′ with word ω′ and indexing set word J2′ J1′ . Then
( a ) ω ≡ G a n d ω ′ = θ˜ 1 ω ≡ θ 1 H w i t h G a n d H i n t h e i n t e r v a l [ F , K ] i n the chain of cyclages on the SSYT’s of weight (q + r, q + s).
( b ) J 2′ J 1′ = Θ˜ J 2 J 1 .
When F = K, we have G = K = H and we recover Theorem 4.3.
Example 4.10. Let F = 2 ← Y = 2 2 and 1112 111
θ1F = 2 ← θ1Y = 2 2 be cyclage chains. The in- 1122 112
dexing sets of F and θ1F are, respectively, F1 = {1, 2, 3}, F2 = {1, 4} and F1′ = {1, 2}, F2′ = {1, 3, 4}. Let U = P54321T15(p), V = P(15)T15(p), and W = P32541T15(p) unimodular matrices, where P54321 and P32541 are the permu- tation matrices associated with 54321 and 32541 ∈ S5 respectively, and T15(p) is the elementary matrix obtained from the identity by placing the prime p in position (3, 5). Let μ = (2, 1, 1).
(1) The sequences ∆μ,UDF and ∆μ,UDθ1F are, respectively, matrix re- alizations for
and
2 T=•12
•••11
2
T′=•22 .
•••11
ThewordswofT andw′ ofT′ satisfyw=21211≡Y andw′ =22211≡θ1Y satisfy θ˜1w = w′, where θ˜1 is the operation based on the parentheses matching (21(21)1).
(2) The sequences ∆μ , V DF and ∆μ , V Dθ1 F are, respectively, matrix re- 22
alizationsforT=• 1 1 andT′=• 1 2 .Wehaveforthe •••12 •••12
wordswofT andw′ ofT′,w=21112≡F andw′ =21212=θ1Y =θ˜1ω̸≡θω based on the parentheses matching (2111)2.
(3) The sequences ∆μ,WDF and ∆μ,WDθ1F are, respectively, matrix re- 22
alizationsforT = • 1 1 andT′ = • 1 1 .Wehave •••12 •••22
w=21112=F and w′ =21122=θ1F.