Page 21 - Textos de Matemática Vol. 39

JEU DE TAQUIN, INVARIANT FACTORS 11
shape, the B-K involution tk and the pairing L-S involution θk agree [19, 22]. The array (2.3) either is always of the form k+1 k+1 or
k+1 k+1
k k k+1k+1.
The Schu¨tzenberger involution evac, based on the evacuation operation, is an involution on SSYT’s which preserves the shape and reverses the weight. It is shown in [17] that it can be defined as evac = t1 t2t1 t3t2t1 ...tn−1 ...t1. Thus evacuation equals the involution τ := θ1 θ2θ1 θ3θ2θ1 . . . θn−1 . . . θ1 on key tableaux. These agreements do not follow for general SSYT’s (see also [17, 21]). Another advantage of the involutions θi is their application to prove that the Littlewood-Richardson number cλμ ν , the number of LR tableaux (tableaux whose rectification is a Yamanouchi tableau ) of shape λ/μ and weight ν, is
independent of the permutation of ν. In particular, the involution τ can be ′
used to exhibit the symmetry cλ′ ′ = cλ where μ′, ν′, λ′ denote the conjugate
μνμν
partitions (see [6]).
3. Action of the symmetric group on the set of k-column words
and on the free algebra
3.1.ColumnwordsandSSYT’s.Givenk≥0,andui ∈V,1≤i≤k,
we define a k-column word as u = (u1,...,uk) ∈ Vk. Two k-columns words
(u1,...,uk) and (v1,...,vk) are equal if ui = vi, 1 ≤ i ≤ k. The shape of
the k-column word u = (u1,...,uk) ∈ V k is (|u1|,...,|uk|). (When the column
word is a SSYT of partition shape, it will be clear from the context whether the
column or row shape is considered.) We shall write often u = u1 · · · uk keeping in
mind that the ui’s are in V . A k-column word u may be identified with a unique
SSYT, with k columns of lengths |u1|, . . . , |uk| and reading word u1 · · · uk, as
follows. For k = 0, the SSYT is ∅, and, for k = 1, is u1. For k = 2, the pair
(u1, u2) is aligned to form a SSYT of two columns with maximal row overlaping
q. This SSYT with reading word u1u2 has shape (q + r1 + s2, q + r1)/(r1, 0),
where q+r1 = |u2| and q+s2 = |u1| for some r1,s2 ≥ 0. For k > 2, each
pair (ui−1,ui) of successive columns of u is aligned to form a SSYT with two
columns as in case k = 2. A frank word is a distinguished column word whose
shape is a permutation of the column shape of the unique SSYT in its plactic
class. In particular, if ui−1ui is a frank word [19, 14], the pair is aligned at: the
top whenever the right column ui is longer, and, in this case, si = 0; and at
the bottom whenever the left column ui−1 is longer, and, in this case, ri−1 = 0.
Note that the outcome SSYT has inner shape ( k−1 rj,...,rk−1,0). A SSYT j=1
satisfying this property for each successive pair of columns is said to be in the compact form. In particular, a SSYT of partition shape is always in the compact form.
kkkkk