Page 30 - Textos de Matemática Vol. 39

20 OLGA AZENHAS
We reduce the study of the invariant factors associated with the sequence of matrices A0, B1, · · · , Bt, satisfying conditions (I) − (III) of Definition 4.1, to the cases ∆μ,UDK and ∆μ,UDF where U is an n by n unimodular matrix, DK denotesthesequence(D[m1],···,D[mt])whichrealizesthekeyKofweight m=(m1,···,mt),andDF denotesthesequence(DF1,···,DFt)whichrealizes the SSYT F with indexing sets F1, · · · , Ft. The combinatorics associated with the sequences ∆μ,UDK involves frank words and words congruent with keys and has been developed in [2, 3, 4, 5, 6] and more recently with R. Mamede in [7, 8, 9]. In this case, the matrix interpretation of the operations θ˜ and Θ˜ and their relationship is as follows. i
Theorem 4.3. [4, 7, 8] Let T and T ′ be the SSYT’s realized by the sequences ∆μ,UDK and ∆μ,UDθ1K, with K the key of weight (m1,m2). Let J2J1, J2′J1′ be the two-column indexing sets, and ω, ω′ the words of T and T′ respectively. Then,
(a) J2J1, J2′ J1′ are frank words such that Θ˜ J2J1 = J2′ J1′ .
( b ) ω ≡ K a n d ω ′ = θ˜ 1 ω ≡ θ 1 K .
(c) there exists an unimodular matrix U ′ such that ∆μ , U ′ DK and
∆μ,U′Dθ1K realize the SSYT’s T and T′′ such that the two-column word of indexing sets J′′J′′ and the word ω′′ of T′′ satisfy J′′J′′ = ΘJ J and ω′′ = θ ω.
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Theorem 4.4. [8] Let T be the SSYT realized by ∆μ,U DK, with word ω and J the column word of indexing sets. Then P (ω) = K and J is a frank word of shape m#.
Theorem 4.5. [7, 9] Let σ ∈< s1,s2 > and θ ∈< θ1,θ2 > with the same reduced word. Let σT be the tableau realized by ∆μ, UDθK with word σω and indexing set column word σJ. Then {σT : σ ∈< s1,s2 >} are the vertices of a hexagon such that
(a)sω=θ˜w≡θω,1≤i≤2,where<θ˜,θ˜ >satisfytheMoore- iii 12
Coxeter relations of the symmetric group S3.
(b) siJ = Θ˜iJ, 1 ≤ i ≤ 2, where < Θ˜1,Θ˜2 > satisfy the Moore-Coxeter
relations of the symmetric group S3.
A complete description of the hexagons defined in (a) and (b) is given in [7, 9]. This family of hexagons contains in particular the hexagon defined by the operators θi and its description is based on the characterization of the Knuth class of a key tableau, over a three-letter alphabet, as the set of the shuffles of its columns [8].
Example 4.6. [9] Let U = P4321T14(p), where P4321 is the permutation ma- trix associated with 4321 ∈ S4 and T14(p) is the elementary matrix obtained from the identity by placing the prime p in position (1, 4). With μ = (2, 1) the
t−i