Ribbon Schur operators

A new combinatorial approach to the ribbon tableaux generating functions and q -Littlewood–Richardson coefficients of Lascoux et al. [A. Lascoux, B. Leclerc, J.-Y. Thibon, Ribbon tableaux, Hall–Littlewood symmetric functions, quantum affine algebras, and unipotent varieties, J. Math. Phys. 38 (3) (1997) 1041–1068] is suggested. We define operators which add ribbons to partitions and following Fomin and Greene [S. Fomin, C. Greene, Noncommutative Schur functions and their applications, Discrete Math. 193 (1998) 179–200] study non-commutative symmetric functions in these operators. This allows us to give combinatorial interpretations for some (skew) q -Littlewood–Richardson coefficients whose non-negativity appears not to be known. Our set-up also leads to a new proof of the action of the Heisenberg algebra on the Fock space of U q ( s l ̂ n ) due to Kashiwara et al. [M. Kashiwara, T. Miwa, E. Stern, Decomposition of q -deformed Fock spaces, Selecta Math. 1 (1996) 787–805].