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 Schutzenberger from C. R. Acad. Sci. Paris Ser. 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 Schutzenberger (C. R. Acad. Sci. Paris Ser. 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.
['Witold Kraśkiewicz', 'Piotr Pragacz']
Schubert functors and Schubert polynomials