Filtrations and recursions for Schubert modules

Revisiting Kraśkiewicz and Pragacz’s construction of Schubert modules, we provide a new proof that their characters are equal to Schubert polynomials. The main innovation is a representation-theoretic interpretation of a recurrence relation for Schubert polynomials recently discovered by Nadeau, Spink, and Tewari. Along the way, we review several related constructions, and show that the Nadeau-Spink-Tewari recursion determines the characters of flagged Schur modules coming from the broader classes of “transparent” and “translucent” diagrams. We conclude with a conjecture concerning the Schubert positivity of the characters of transparent diagrams.

💡 Research Summary

The paper revisits the construction of Schubert modules introduced by Kraśkiewicz and Pragacz and provides a new proof that the characters of these modules coincide with Schubert polynomials. The central novelty is a representation‑theoretic interpretation of a recursion for Schubert polynomials discovered recently by Nadeau, Spink, and Tewari. This recursion involves the Bergeron‑Sottile operators (R_k), which act on a polynomial by sending (x_k) to zero and shifting all variables with index larger than (k) down by one. The authors show that these operators arise naturally from a filtration of the Kraśkiewicz‑Pragacz modules (E_D(w)^\bullet).

The filtration is built as a chain \