Schubert polynomials of types A-D
Schubert polynomials are explicit representatives for Schubert classes in the cohomology ring of a flag variety. Those of type \(A_n\) were introduced by \textit{A. Lascoux} and \textit{M. P. Schürzenberger} [Polynomes de Schubert, C. R. Acad. Sci. Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)]. \textit{S. Billey} and \textit{M. Haiman} [Schubert polynomials for the classical groups, J. Am. Math. Soc. 8, No. 2, 443-482 (1995; Zbl 0832.05098)] extended the theory of \(A_n\)-Schubert polynomials for the groups of type \(B_n, C_n\) and \(D_n\) and their flag varieties using combinatorial methods. The starting point for the theory of Schubert polynomials is the observation of \textit{I. N. Bernstein, I. M. Gelfand} and \textit{S. I. Gelfand }[Schubert cells and cohomology of the spaces \(G/P\), Russ. Math. Surveys 28, No. 3, 1-26 (1973; Zbl 0286.57025)] that all Schubert classes can be computed by applying a sequence of divided difference operators to the cohomology class of highest codimension (the Schubert class of a point). For the type \(A_n\) Lascoux and Schürzenberger found a particular polynomial to represent the top cohomology class which yields Schubert polynomials that represent the Schubert classes simultaneously for all \(n\), the top polynomial. Billey and Haiman [loc. cit.] described the top polynomials of type \(B_n, C_n\) and \(D_n\). In the paper under review the author follows closely the original algebraic approach of Lascoux and Schürzenberger in type \(A_n\). He is able to present simple formulas for the top polynomials of type \(C_n\) and \(D_n\). He uses creation operators for \(Q\)-Schur and \(P\)-Schur functions which also allows him in types \(B_n, C_n\) and \(D_n\) to give: (1) formulas for the easy computation with all divided differences, (2) recursive structures, and (3) simplified derivations of basic properties.
scientific article; zbMATH DE number 1355506