Saturation property fails for Schubert coefficients

TL;DR

This paper disproves the conjecture that the saturation property extends from Littlewood--Richardson to Schubert coefficients, showing it fails in many cases and discussing related computational complexity issues.

Contribution

It provides the first strong disproof of the saturation property for Schubert coefficients and explores its failure under bit scaling.

Findings

Saturation property fails for a large family of Schubert coefficient instances

Disproves Kirillov's conjecture on saturation for Schubert coefficients

Discusses computational complexity implications of the failure

Abstract

The saturation property for Littlewood--Richardson coefficients was established by Knutson and Tao in 1999. In 2004, Kirillov conjectured that the saturation property extends to Schubert coefficients. We disprove this conjecture in a strong form, by showing that it fails for a large family of instances. We also refute the saturation property for Schubert coefficients under bit scaling and discuss computational complexity implications.