Kra\'skiewicz-Pragacz modules and Pieri and dual Pieri rules for Schubert polynomials

TL;DR

This paper constructs explicit filtrations of tensor products of KP modules, which have characters as Schubert polynomials, focusing on cases related to Pieri and dual Pieri rules, advancing understanding of their algebraic structure.

Contribution

It provides explicit KP filtrations for tensor products involving Schubert modules and symmetric or exterior powers, specifically in Pieri and dual Pieri cases.

Findings

Explicit KP filtrations constructed for specific tensor products.

Demonstrates the structure of tensor products related to Pieri rules.

Advances algebraic understanding of Schubert polynomial modules.

Abstract

In their 1987 paper Kra\'skiewicz and Pragacz defined certain modules, which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of KP modules always has a KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely $\mathcal{S}_w \otimes S^d(K^i)$ and $\mathcal{S}_w \otimes \bigwedge^d(K^i)$, corresponding to Pieri and dual Pieri rules for Schubert polynomials.