Séminaire Lotharingien de Combinatoire, B53h (2006), 22 pp.
Olga Azenhas and Ricardo Mamede
Matrix Realizations of Pairs of Young Tableaux, Keys
and Shuffles
Abstract.
A key H 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-letters
alphabet the Knuth class of a key equals the set of shuffles of its
columns, and on a four-letters 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-Sch\"utzenberger
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-letters alphabet, and they are used
to show that,
over a t-letters alphabet, the pair (T,H) has a matrix
realization only if the word of T is in the Knuth class of
H.
Received: December 22, 2004.
Revised: July 24, 2006.
Accepted: August 14, 2006.
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