Actions of the symmetric group on sets generated by Yamanouchi words

Abstract

In this paper we consider words in a finite totally ordered alphabet which, restricted to a two consecutive letters subalphabet, are either Yamanouchi or dual Yamanouchi. We introduce coordinates or indexing sets of words and we show that there is a monoid isomorphism between words and classes of sequences of finite sets in N. Considering words in a two consecutive letters subalphabet, we define maps acting on pairs of indexing sets which, by fixing a longest self-dual Yamanouchi subword, transform a Yamanouchi into a dual Yamanouchi word, and reciprocally. The pairs of indexing sets of Yamanouchi and dual Yamanouchi words are, respectively, comparable under an ordering and its dual in the power-set of {1,...,n}. This family of transformations is induced by the witnesses of the comparable pairs. When minimal and maximal witnesses are considered, we recover those operators which satisfy the conditions of the symmetric group, defined by A. Lascoux and M. P. Schutzenberger. Starting with given indexing sets of a Yamanouchi word, in a three-letters alphabet, we generate, under the action of these transformations, a set of indexing sets which gives rise to an action of the symmetric group S_3. This group action of S_3 is equivalent to an explicit decomposition of the given indexing sets of a Yamanouchi word in a three-letters alphabet. For transformations induced by minimal and maximal witnesses, we use this decomposition to define, recursively, an action of the symmetric group S_3, for t greater or equal to 3, on a set generated by indexing sets of all Yamanouchi words in a t-letters alphabet. This group action coincides with the one described by A. Lascoux and M. P. Schutzenberger, when restricted to the words under consideration. The action of the symmetric group S_3, on words or Young tableaux, has a natural matrix translation afforded by the obvious permutation action on a sequence of matrices over a local principal ideal domain with maximal ideal (p). Moreover, such a permutation action gives rise, directly, to the mentioned decomposition of the indexing sets of a Yamanouchi word in a three-letters alphabet. This is the content of a subsequent paper.