Schubert functors and Schubert polynomials

We construct a family of functors assigning an R-module to a flag of R-modules, where R is a commutative ring. As particular instances, we get flagged Schur functors and Schubert functors, the latter family being indexed by permutations. We identify Schubert functors for vexillary permutations with some flagged Schur functors, thus establishing a functorial analogue of a theorem of Lascoux and Schützenberger from C. R. Acad. Sci. Paris Sér. I Math. 294 (1982) 447 and of Wachs from J. Combin. Theory Ser. A 40 (1985) 276. Over an infinite field, we study the trace of a Schubert module, which is a cyclic module over a Borel subgroup B, restricted to the maximal torus. The main result of the paper says that this trace is equal to the corresponding Schubert polynomial of Lascoux and Schützenberger (C. R. Acad. Sci. Paris Sér. I Math. 294 (1982) 447). We also investigate filtrations of B-modules associated with the Monk formula (Proc. London Math. Soc. 9 (1959) 253) and transition formula from Lett. Math. Phys. 10 (1985) 111.