Page 18 - Textos de Matemática Vol. 39

8 OLGA AZENHAS
class. This action on k-column words is everywhere defined and in turn extends the one in [19], on k-column words, not everywhere defined. On the other hand, in the free algebra, that is, on all words, this action is translated into to the Lascoux-Schu¨tzenberger action of the symmetric group based on the standard pairing of parentheses [18, 22].
In Section 2, the combinatorial definition of Schur function where the Bender-Knuth involution plays an important role is reviewed as well as its relationship with other involutions on semi-standard Young tableaux [17, 21, 24]. In Section 3, we extend the action of the symmetric group defined by Lascoux-Schu¨tzenberger on frank words [19] to arbitrary k-column words in a plactic class; the translation to the action of the symmetric group in the free algebra via a variant of the dual RSK-correspondence is explained. In the last section, the previous combinatorial operations are interpreted in the context of the invariant factors of matrices over a local principal ideal domain, and, in this context, some interesting generalizations arise. The combinatorics of the invariant factors and its relationship with Yamanouchi tableaux (Littlewood- Richardson tableaux of partition shape) has been developed earlier by several authors, like J. A. Green, T. Klein and R. C. Thompson et al in [12, 25, 2, 3, 6, 1], with key-tableaux and frank words in [3, 4, 5] and, more recently, in [7, 8, 9], with R. Mamede.
2. Schur functions and semi-standard Young tableaux
2.1. Symmetric functions. Let N be the set of nonnegative integers. Let x = (x1,x2,...) be a vector of indeterminates and n ∈ N. A homogeneous symmetric function of degree n over a commutative ring R with identity is a formal power series f(x) = α cαxα where α runs over all weak compositions α=(α1,α2,···)ofn,cα ∈Randxα standsforthemonomialxα1xα2 ···,such that f(xπ(1),xπ(2),...) = f(x1,x2,...) for every permutation π of the positive integers [24]. The set of all homogeneous symmetric functions of degree n over R is an R-module and a vector space when R = Q. Different bases for this vector space are known. An important one is given by the monomial symmetric functions
(2.1) mλ =xα α
with λ a partition of n, written λ| − n, and α running over all distinct permu- tations of the entries of λ.
2.2. Semi-standard Young tableaux. Given t ≥ 1, we put [t] := {1, . . . , t} and denote by [t]∗ the free monoid in the alphabet [t] and by ε the empty word. The word ω = x1 · · · xr over the alphabet [t] is called a column word if x1 >···>xr,andarowwordifx1 ≤x2 ≤···≤xr.Thesetofcolumnsof