When do Schubert polynomial products stabilize?

TL;DR

This paper provides an explicit formula for the back-stabilization number of Schubert polynomial products, proving a conjecture and offering new combinatorial insights into their stabilization behavior.

Contribution

It introduces a formula for the back-stabilization number and the minimal n for complete Schubert product expansion, using combinatorial methods involving colored words and differential operators.

Findings

Explicit formula for back-stabilization number

Proof of N. Li's conjecture in a strengthened form

New combinatorial tools for analyzing Schubert polynomial products

Abstract

The "back-stabilization number" for products of Schubert polynomials is the distance the corresponding permutations must be shifted before the structure constants stabilize. We give an explicit formula for this number and thereby prove a conjecture of N. Li in a strengthened form. This leads to an additional result: a formula for the smallest $n$ such that a given Schubert product expands completely over $S_n$. Our method is to explore back-stable fundamental slide polynomials and their products combinatorially, in the context of their associated words. We use three main tools: (i) an algebra consisting of "colored words", with a modified shuffle product, and which contains the rings of back (quasi)symmetric functions as subquotients; (ii) the combinatorics of increasing suffixes of reduced words; and (iii) the lift of differential operators to the space of colored words.