# Mathematics Colloquia and Seminars

Ribbon tableaux, or rim-hook tableux, are generalized Young tableaux which are used to define a family of symmetric functions which are $q$ analogues of products of Schur functions. When these functions were introduced by Lascoux, Leclerc and Thibon, a geometric proof of symmetry was given using Fock space representations, but no Schur-positivity results were known. Since then, several proofs, some more sound than others, have given a combinatorial description in terms of Schur functions for the case of dominoes (2-ribbons), but higher cases still remain unsolved. In this talk, we will present a new combinatorial proof of the domino case modeled after the classical proof of the Littlewood-Richardson rule. The advantage of this new proof is that, unlike other solutions, it has a natural, albeit difficult, generalization to 3-ribbons and possibly beyond.