Given a pair of tableaux (T ;K(¾)), where T is a skew-tableau in thealphabet [t] and K(¾) is the key associated with ¾ 2 St, with the same evaluationas T , we consider the problem of a matrix realization for (T ;K(¾)) over a localprincipal ideal domain [1, 2, 3, 4, 5, 6]. It has been shown that the pair (T ;K(¾))has a matrix realization only if the word of T is in the plactic class of K(¾) [5].This condition has also been proved su±cient when ¾ is the identity [1, 2, 4], thereverse permutation in St [2, 3], or any permutation in S3 [6]. In each of these cases,the plactic class of K(¾) may be described by shu²ing together their columns. Fort ¸ 4 this is no longer true for an arbitrary permutation, but shu²ing together thecolumns of a key always leads to a congruent word. In [17] A. Lascoux and M. P.SchÄutzenberger have introduced the notions of frank word and key. It is a simplederivation on Greene's theorem [11] that words congruent with a key, and frankwords are dual of each other as biwords. In this paper, we exhibit, for any ¾ 2 St,a matrix realization for the pair (T ;K(¾)), when the word of T is a shu²e of thecolumns of K(¾). This construction is based on a biword de¯ned by the columns ofthe key and the places of their letters in the skew-tableau T . The places of theseletters are row words which are shuffle components of a frank word.
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