On the expansion of Schur and Schubert polynomials into standard elementary polynomials

摘要

Motivated by the recent discovery of a simple quantization procedure for Schubert polynomials we study the expansion of Schur and Schubert polynomials into standard elementary monomials (SEM). The SEM expansion of Schur polynomials can be described algebraically by a simple variant of the Jacobi-Trudi formula and combinatorially by a rule based on posers of staircase box diagrams. These posits are seen to be rank symmetric and order isomorphic to certain principal order ideals in the Bruhat order of symmetric groups ranging between the full sym metric group and the respective maximal Boolean sublattice. We prove and conjecture extensions of these results for general Schubert polynomials. The featured conjectures are: (1) an interpretation of SEM expansions as "alternating approximations" and (2) surprising properties of different numbers naturally associated to SEM expansions. This hints at as yet undiscovered deeper symmetry properties of the SEM expansion of Schubert polynomials. (C) 1998 Academic Press. [References: 18]