Matrix Realizations of Pairs of Young Tableaux, Keys and Shuffles

A keyH is a semi-standard tableau of partition shape whose evaluation is a permutation of the shape. We give a necessary and sufficient condition that the Knuth class of a key equals the set of shuffles of its columns. In particular, on a three-letter alphabet the Knuth class of a key equals the set of shuffles of its columns, and on a four-letter alphabet, the Knuth class of a key is either the set of shuffles of its columns or the set of shuffles of its distinct columns with a single word taking appropriate multiplicities. For some instances of H this result has been already applied to exhibit a matrix realization, over a local principal ideal domain, of a pair of tableaux (T ,H), where T is a skew-tableau whose word is in the Knuth class of H. Generalized Lascoux–Schützenberger operators, based on nonstandard matching of parentheses, arise in the matrix realization, over local principal ideal domain, of a pair (T ,H) on a two-letter alphabet, and they are used to show that, over a t-letter alphabet, the pair (T ,H) has a matrix realization only if the word of T is in the Knuth class of H.