Embedding bumpless pipedreams as Bruhat chains

TL;DR

This paper explores combinatorial formulas for Schubert polynomials using Bruhat order, establishing a new connection between bumpless pipedreams and classical pipedreams, with applications to algebraic structures and bijections.

Contribution

It introduces a novel perspective linking bumpless pipedreams to Bruhat chains, extending classical pipedream formulas and providing new algebraic and combinatorial insights.

Findings

Bumpless pipedreams correspond to an extreme case of Bruhat chain formulas.

A bijection between pipedreams and bumpless pipedreams is established via growth diagrams.

The approach offers algebraic and combinatorial tools for studying Schubert polynomials.

Abstract

Schubert polynomials are distinguished representatives of Schubert cycles in the cohomology of the flag variety. In the spirit of Bergeron and Sottile, we use the Bruhat order to give $(n-1)!$ different combinatorial formulas for the Schubert polynomial of a permutation in $S_n$. By work of Lenart and Sottile, one extreme of the formulas recover the classical Pipedream (PD) formula. We prove the other extreme corresponds to Bumpless pipedreams (BPDs). We give two applications of this perspective to view BPDs: Using the Fomin-Kirrilov algebra, we solve the problem of finding a BPD analogue of Fomin and Stanley's algebraic construction on PDs; We also establish a bijection between PDs and BPDs using Lenart's growth diagram, which conjectually agrees with the existing bijection of Gao and Huang.