Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties

We introduce a new family of symmetric functions, which are q analogs of products of Schur functions, defined in terms of ribbon tableaux, These functions can be interpreted in terms of the Fock space representation F-q of U-q(<(sl(n))over cap>), and are related to Hall-Littlewood functions via the geometry of flag varieties. We present a series of conjectures, and prove them in special cases, The essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of F-q using ribbon tableaux. (C) 1997 American Institute of Physics.