SPLITTING THE SQUARE OF A SCHUR FUNCTION INTO ITS SYMMETRICAL AND ANTISYMMETRIC PARTS

We propose a new combinatorial description of the product of two Schur functions. In the particular case of the square of a Schur function SI, it allows to discriminate in a very natural way between the symmetric and antisymmetric parts of the square. In other words, it describes at the same time the expansion on the basis of Schur functions of the plethysms S-2(S-I) and Lambda(2)(S-I). More generally our combinatorial interpretation of the multiplicities c(IJ)(K) = (SISJ, S-K) leads to interesting q-analogues C-IJ(K)(q) Of these multiplicities. The combinatorial objects that we use are domino tableaux, namely tableaux made up of 1 x 2 rectangular boxes filled with integers weakly increasing along the rows and strictly increasing along the columns. Standard domino tableaux have already been considered by many authors [33], [6], [34], [8], [1], but, to the best of our knowledge, the expression of the Littlewood-Richardson coefficients in terms of Yamanouchi domino tableaux is new, as well as the bijection described in Section 7, and the notion of the diagonal class of a domino tableau, defined in Section 8. This construction leads to the definition of a new family of symmetric functions (H-functions), whose relevant properties are summarized in Section 9.