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Author(s)
Olga Azenhas; Ricardo Mamede;
Title Matrix realization of a pair of tableaux with key and shuffling condition
Abstract Given a pair of tableaux (T,K(\sigma)), where T is a skewtableau in the alphabet [t] and K(\sigma) is the key associated with \sigma\in S_{t}, with the same evaluation as T, we consider the problem of a matrix realization for (T,K(\sigma)) over a local principal ideal domain [1, 2, 3, 4, 5, 6]. It has been shown that the pair (T,K(\sigma)) has a matrix realization only if the word of T is in the plactic class of K(\sigma) [5].
This condition has also been proved suffcient when \sigma is the identity [1, 2, 4], the reverse permutation in S_{t} [2, 3], or any permutation in S_{3} [6]. In each of these cases, the plactic class of K(\sigma) may be described by shuffling together their columns. For
t\geq 4 this is no longer true for an arbitrary permutation, but shuffling together the columns of a key always leads to a congruent word. In [17] A. Lascoux and M. P. Schutzenberger 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 \sigma\in S_{t},
a matrix realization for the pair (T,K(\sigma)), when the word of T is a shuffle of the columns of K(\sigma). This construction is based on a biword defined by the columns of the key and the places of their letters in the skewtableau T. The places of these letters are row words which are shuffle components of a frank word.
Preprint series Prépublicações do Departamento de Matemática da Universidade de Coimbra
Issue 0525
Year 2005

