On the expansion of Schur and Schubert polynomials into standard elementary monomials
Any polynomial in \(\mathbb{Z}[x]\) can be expressed in terms of elementary symmetric polynomials which can in turn be expressed as linear combinations of standard elementary monomials or SEM. The paper under review examines the SEM expansions of Schur and Schubert polynomials. The SEM expansion for Schubert polynomials is of particular interest because quantum Schubert polynomials can be computed by quantizing the SEM expansion of ordinary Schubert polynomials [\textit{S. Fomin}, \textit{S. Gelfand} and \textit{A. Postnikov}, Quantum Schubert polynomials, J. Am. Math. Soc. 10, No. 3, 565-596 (1997; Zbl 0912.14018)]. In the case of Schur functions, the SEM expansion can be obtained via a variant of the Jacobi-Trudi identity, and the author demonstrates how a combinatorial rule based on posets of staircase box diagrams can be used to calculate SEM expansion coefficients. In the case of Schubert polynomials, the SEM expansion can be obtained via the determinantal expression of \textit{A. N. Kirillov} and \textit{T. Maeno} [Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula, q-alg/9610022, 1996, preprint, 52 pp.] and the author conjectures a combinatorial rule similar to that proved for Schur functions.
scientific article; zbMATH DE number 1215516
scientific article