Séminaire Lotharingien de Combinatoire, B57f (2010), 38 pp.

Olga Azenhas and Ricardo Mamede

Matrix Tableau-Pairs with Key and Shuffling Conditions

Abstract. It has been shown that the sequence of Smith invariants defined by certain sequences of products of matrices, with entries in a local principal ideal domain, are combinatorially described by tableau-pairs (T,K) where T is a tableau of skew-shape which rectifies to the key-tableau K. It is a fact that the set of all shuffles of the columns of a key-tableau is a subset of its Knuth class. Here, under the condition that the word of T is a shuffle of the columns of the key-tableau K, we show the converse, that is, every tableau-pair under the aforesaid restrictions has a matrix construction. In the case of a four-letter alphabet, since we are able to give an explicit description of the Knuth class of a key-tableau as a union of the shuffles of certain subsets of words containing the key-tableau columns, our construction is general. This may be seen as an indication of a general procedure if a subset of shuffling generators of a generic key-tableau Knuth class is provided. At the moment however, this seems to be a very difficult problem.

Received: April 23, 2007. Revised: November 1, 2010. Accepted: November 12, 2010.

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