Page 20 - Textos de Matemática Vol. 39

10 OLGA AZENHAS
For instance for the SSYT’s above (2.2) we have xT = x31x2x3x4x25 and xH = x3x24x35x6.
Given μ ⊆ λ, the skew-Schur function sλ/μ of shape λ/μ in the variables x1, x2, . . . is the formal power series Sλ/μ(x) = T xT where T runs over all SSYT’s of shape λ/μ. If μ = ∅, λ/μ = λ and we call sλ(x) the Schur function of shape λ.
Theorem 2.1. [10] The number of SSYT’s of shape λ/μ with weight α is independent of the permutations of the entries of α.
Proof. The Bender-Knuth involution tk, for short B-K, on the set SSYT’s of shape λ/μ and weight α, introduced in [10], performs an interchange of the contiguous components αk and αk+1 in the weight α and leaves the shape of T unchanged. Let T be a SSYT of shape λ/μ and weight α. All entries tij ̸= k,k+1 remain unchanged. A portion of T with parts equal to k or k+1 has the form
k+1 k+1
(2.3) k k k k k k+1 k+1 k+1 k+1 k+1
k k k.
Ignoring the columns with both parts k and k + 1, we obtain in each row of T a word of the form kr(k + 1)s, for some r, s ≥ 0, which we replace with ks(k + 1)r. The transformation tk acts on T by performing this interchange, independently for each row, leaving the column words k + 1 k unchanged.
From this combinatorial result it follows
Corollary 2.2. The skew-Schur functions are symmetric functions.
Clearly the B-K involution tk satisfies tkti = titk for |i − k| > 2 but does not give rise to an action of the symmetric group on SSYT’s. As one may check, the application of t1t2t1 and t2t1t2 to T above (2.2) leads to different results, although both have the same weight (1, 1, 3, 4, 2, 1, 0, 1). In the proof of Theorem 2.1 we might have used the Lascoux-Schu¨tzenberger involution θk, for short L-S involution, which satisfies the Moore-Coxeter relations of the symmetric group. The involution θk, based on the standard parenthesis pairing procedure, is described as follows. Let ω ∈ [t]∗. First we extract from ω a subword ω′ containing the letters k and k + 1 only. Second, in the word ω′ we remove consecutively all factors k + 1 k. As a result we obtain a subword of the form kr k + 1s. We replace it with the word ks k + 1r and, after this, recover all the removed pairs and all the letters which differ from k and k + 1. We arrive to a new word, denoted by θkω in [t]∗ whose weight is the interchange of (wtω)k with (wtω)k+1. In general, tk ̸= θk, as may be easily checked applying θ2 and t2 to T (2.2).
Nevertheless it is worth to point out that over the key tableaux [14, 19], that is SSYT’s of partition shape whose weight is a permutation of the