Page 19 - Textos de Matemática Vol. 39

JEU DE TAQUIN, INVARIANT FACTORS 9
[t]∗ is denoted by V . The length r of ω is written |ω|. We define the weight of ω ∈ [t]∗ as wtω ∈ Nt, where (wtω)i counts the number of letters i in ω.
Given a partition λ, the number of its nonzero parts l(λ) is the length of λ. A partition is identified with the diagram of the boxes arranged in left justified rows at the bottom. (The French convention is adopted, that is, the longest row of the partition is in the bottom.) The empty partition is denoted by ∅. A semi-standard Young tableau, for short SSYT, of shape λ is an array T =(tij),1≤i≤l(λ),1≤j≤λi,ofpositiveintegersofshapeλ,weakly increasing in every row and strictly decreasing down in every column. Let μ be a partition such that μ ⊆ λ, that is, μi ≤ λi. The diagram λ/μ, obtained from λ by removing μ, is called a skew-diagram. Similarly, we define a semi-standard Young tableau of shape λ/μ as an a array T = (tij), 1 ≤ i ≤ l(λ), μi < j ≤ λi. The partitions λ and μ are called, respectively, the outer and the inner shape of the semi-standard Young tableau of shape λ/μ. When μ = ∅ we get a semi- standard Young tableau of partition shape. Examples of semi-standard Young tableaux of shape (5, 4, 2) and (4, 4, 2, 1)/(3, 1) are
5 5 65 5
(2.2) T=2444 andH=445.
11134 3
The reading word of a semi-standard Young tableau T is the sequence of entries of T obtained by concatenating the column words of T left to right. For instance the reading word of T is 52154141434 and of H is 6554453. The weight of a SSYT is the weight of its reading word. T has weight (3, 1, 1, 4, 2). A SSYT of partition shape is identified with its reading word.
A SSYT T of shape λ/μ and weight α may also be represented by a nested sequence of partitions T = (λ0,λ1,...,λt), where μ = λ0 ⊆ λ1 ⊆ ··· ⊆ λt = λ, such that for k = 1, . . . , t, the skew diagram λk/λk−1 is labeled by k, with αk = |λk| − |λk−1| [24].
Cyclic permutations on words induce an operation on SSYT’s of partition shape, called cyclage, providing a rank-poset structure on the of SSYT’s of given weight α; the row word 1α1 2α2 . . . is the unique minimal element [18, 20]. For instance, the cyclage chain of the SSYT’s with weight (3, 2) is the interval [1322, Y = 21211], defined by
Y=22 2
1 1 1 → 1 1 1 2→ 1 1 1 2 2.
2.3. Schur functions. There are several ways to define Schur functions [24]. For our purposes, we adopt the one in terms of mλ (2.1) which exhibits the relationship with semi-standard Young tableaux. For any semi-standard Young
tableauT ofweightα=(α1,α2,···),wedefinethemonomialxT =xα1xα2 .... 12