Page 17 - Textos de Matemática Vol. 39

SCHUR FUNCTIONS, PAIRING OF PARENTHESES, JEU DE TAQUIN AND INVARIANT FACTORS
OLGA AZENHAS
Dedicated to Eduardo Marques de Sa´ on the occasion of his 60th birthday.
Abstract. We translate the coplactic operation by Lascoux and Schu¨tzenber- ger which, based on the standard pairing of parentheses, transforms a word on a two-letter alphabet into one of reversed weight ([18, 22]), to the action of the jeu de taquin on two neighbor columns of a semi-standard Young tableau of skew shape. That enables to extend the action of the symmetric group on frank words to arbitrary k-column words, in a Knuth class. On the other hand, considering variants of the jeu de taquin on two-column words, that allows us to introduce variants of the mentioned Lascoux-Schu¨tzenberger operation on words, based on nonstandard pairing of parentheses, and to give a combinatorial description of the invariant factors associated with certain types of sequences of product of matrices, over a local principal ideal domain.
1. Introduction
It is well-known that there is a remarkable relationship between the combi- natorics of semi-standard Young tableaux and Schur functions [14, 23, 24]. Not so well-known is the relationship between those combinatorial objects with the invariant factors of matrices over a local principal ideal domain. Indeed a high- light in this analogy is the fact that the Littlewood-Richardson rule describes the invariant factors of a product of matrices over a local principal ideal domain, as well as the product of two Schur functions as a linear combination of the same functions [14, 15, 16, 24]. In this paper we go further and some important combinatorial operations on semi-standard Young tableaux like Bender-Knuth involution [10], Lascoux-Schu¨tzenberger operators based on standard pairing of parentheses [18, 22], and Schu¨tzenberger jeu de taquin, are interpreted in the context of the invariant factors of matrices over a local principal ideal domain. On the way, we extend the action of the symmetric group defined by Lascoux- Schu¨tzenberger on frank words [19, 14] to arbitrary k-column words in a plactic
2000 Mathematics Subject Classification. 05E05, 05E10, 15A45.
Key words and phrases. Invariant factors, jeu de taquin, Lascoux-Schu¨tzenberger action of the symmetric group on the free algebra, matrices over a local principal ideal domain, pairing of parentheses, and Schur functions.
The work was supported by CMUC/FCT and SFRH/BSAB/515/2005, grant by the Por- tuguese Foundation of Science and Technology, FCT, at the Combinatorics Group, Fakult¨at fu¨r Mathematik, Universit¨at Wien, Austria.
7