Matrix realization of a pair of tableaux with key and shuffling condition

Abstract

Given a pair of tableaux (T ;K(¾)), where T is a skew-tableau in the alphabet [t] and K(¾) is the key associated with ¾ 2 St, with the same evaluation as T , we consider the problem of a matrix realization for (T ;K(¾)) over a local principal 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], the reverse 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. For t ¸ 4 this is no longer true for an arbitrary permutation, but shu²ing together the columns 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 simple derivation on Greene's theorem [11] that words congruent with a key, and frank words 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 the columns of K(¾). This construction is based on a biword de¯ned by the columns of the key and the places of their letters in the skew-tableau T . The places of these letters are row words which are shuffle components of a frank word.