Crystal Chute Moves on Pipe Dreams

TL;DR

This work develops a Demazure crystal framework on reduced pipe dreams that index Schubert polynomials, enabling a decomposition into key polynomials. The authors define crystal chute moves as restricted Billey–Bergeron moves, yielding a Demazure crystal structure on $RP(w)$ and an explicit weight-preserving bijection to RFC$(w^{-1})$, unifying pipe dreams, rc-graphs, and reduced factorizations with cutoff. Consequently, they express $\mathfrak{S}_w$ as a sum of key polynomials $\kappa_a$ indexed by highest-weight pipe dreams, with a computable truncating permutation $\pi_D$ that records each component. The framework links several combinatorial models and employs Edelman–Greene insertion to index Demazure components, providing representation-theoretic insight into Schubert calculus and new avenues for geometric interpretation.

Abstract

Schubert polynomials represent a basis for the cohomology of the complete flag variety and thus play a central role in geometry and combinatorics. In this context, Schubert polynomials are generating functions over various combinatorial objects, such as rc-graphs or reduced pipe dreams. By restricting Bergeron and Billey's chute moves on rc-graphs, we define a Demazure crystal structure on the monomials of a Schubert polynomial. As a consequence, we provide a method for decomposing Schubert polynomials as sums of key polynomials, complementing related work of Assaf and Schilling via reduced factorizations with cutoff, as well as Lenart's coplactic operators on biwords.

Figures (14)

• Figure 1.1: Demazure crystal structure on reduced pipe dreams for $w = [21543]$.
• Figure 2.1: Several reduced pipe dreams for $w = [21543] \in S_5$.
• Figure 2.2: The pairing process applied to row 1 of a reduced pipe dream.
• Figure 2.3: A (crystal) chute move swaps the unpaired cross on the upper right with the elbow on the lower left, preserving the rectangle of crosses in between.
• Figure 2.5: Rectangles of crosses not bordered by elbows are non-reduced.

Theorems & Definitions (75)

• Theorem 1.1
• Theorem 2.1: Corollary 2.1.3 KnutsonMiller
• Example 2.2
• Definition 2.3
• Example 2.4
• Definition 2.5
• Remark 2.6
• Example 2.7
• Proposition 2.8
• proof