Page 31 - Textos de Matemática Vol. 39

1 (a)U=I,• 1
2 (b)U=P4321,• 1
2
(c) U = P12543P4321, • 1
JEU DE TAQUIN, INVARIANT FACTORS 21
sequences ∆μ,UD[3],D[2] and ∆μ,UD[2],D[3] are, respectively, matrix realiza- 22
tionsforT =• 1 2 andT′ =• 2 2 .Thewordsw=21211ofT ••11 ••11
and w′ = 22211 of T′ satisfy θ˜1w = w′ ≡ θiω, where θ˜1 is the operation based on the parentheses matching (21(21)1). However, if we choose U′ = P3241T24(p), with P3241 the permutation matrix associated with 3241, the sequences ∆μ,U′ D[3], D[2] and ∆μ,U′ D[2], D[3] are, respectively, matrix realizations for T and
2
T′′ =• 1 2 .Inthiscase,thewordw′′ ofT′′ satisfyθ1w=w′′.The
••12
corresponding operations on the indexing set words (frank words) are displayed in (4.5).
Given F a two-letter SSYT of partition shape, its indexing sets (F1,F2) are F1 = [q]∪[q+1, q+r] and F2 = [q]∪[q+r+1, q+r+s] for some q, r, s ≥ 0. The indexing sets of θ1F are F1′ = [q]∪[q+1,q+s] and F2′ = [q]∪[q+s+1,q+s+r]. Wheneitherrors=0,F isakeytableauK.
Theorem 4.7. Let ∆μ,UDF be a matrix realization of T with word ω and two-column word of indexing sets J = J2J1. Then
(a) ω ≡ G with G in the interval [F,K] of the chain of cyclages on the SSYT’s of weight (q + r, q + s).
(b) J2J1 is a two-column SSYT, in the compact form, with column lengths (q + s, q + r) and inner shape (r − f ), for some 0 ≤ f ≤ min{r, s}.
Example 4.8. Let μ = (3,2,1) and F = 11122. The sequences ∆μ,UDF, with U running over the unimodular matrices of order 5, give rise to SSYT’s of inner shape μ with words congruent with P running over {11122; 21112; 21 21 1}. For instance,
,ω=11122;
,ω=21112;
, ω = 21211.
••1 •••22
••1 •••12
••2 •••12