The Skew Schubert Polynomials

William Y. C. Chen, Guo-Guang Yan and Arthur L. B. Yang

  Abstract:   We obtain a tableau defnition of the skew Schubert polynomials named by Lascoux, which are defined as a flagged double skew Schur functions. These polynomials are in fact Schubert polynomials in two sets of variables indexed by 321-avoiding permutations. From the divided difference definition of the skew Schubert polynomials, we construct a lattice path interpretation based on the pairing lemma of Chen-Li-Louck. The lattice path explanation immediately leads to the determinantal definition and the tableau definition of the skew Schubert polynomials. For the case of a single variable set, the skew Schubert polynomials reduce to a flagged skew Schur functions as studied by Wachs and Billey-Jockusch-Stanley. We also present a lattice path interpretation of the isobaric divided difference operators, and derive an expression of the a flagged Schur function in terms of isobaric operators acting on a monomial. Moreover, we find lattice path arguments for the Giambelli identity and the Lascoux-Pragacz identity for super Schur functions. For the super Lascoux-Pragacz identity, the lattice path construction is related to the code of a partition which determines the directions of the lines parallel to the y-axis in the lattice.

  Keywords:  Lattice path, isobaric divided difference, a flagged double skew Schur function, skew Schubert polynomial, Giambelli identity, Lascoux-Pragacz identity, key polynomial, code of a partition.

   AMS Classification:
 05E05, 05A15.