Schubert polynomials of types A--D

Abstract:

Schubert polynomials of type B, C, and D have been described first by S. Billey and M. Haiman [BH] using a combinatorial method. In this paper we give a unified algebraic treatment of Schubert polynomials of types A–D in the style of the Lascoux–Schützenberger theory in type A, i.e. Schubert polynomials are generated by the application of sequences of divided difference operators to “top polynomials”. The use of the creation operators for Q-Schur and P-Schur functions allows us to give: (1) simple and natural forms of the “top polynomials”, (2) formulas for the easy computation with all divided differences, (3) recursive structures, and (4) simplified derivations of basic properties.