The Skew Schubert Polynomials
William Y.C. Chen, G.-G. 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. |