Saturation property fails for Schubert coefficients

TL;DR

This work investigates whether the saturation property extends from Littlewood–Richardson coefficients to Schubert coefficients. It provides strong counterexamples to Kirillov's conjecture, showing that there exist triples with $c_{u,v}^w>0$ (in fact $c_{u,v}^w=1$) for which $c_{N*u,N*v}^{N*w}=0$ for all $N>1$, using Monk's rule and the StDizier–Yong vanishing condition. The paper further demonstrates that a natural bit-scaling variant, $N\otimes w$, also destroys saturation in broad families, again with explicit counterexamples. These results have implications for Schubert vanishing problems and computational complexity, indicating that saturation-based LP approaches do not generalize straightforwardly beyond LR coefficients. Overall, the findings delineate the limits of saturation phenomena in Schubert calculus and inform ongoing inquiries into the complexity of Schubert-structure constants.

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.

Theorems & Definitions (18)

• Conjecture 1.1: Kirillov Kir04
• Theorem 1.2
• Corollary 1.3
• Proposition 2.1: Monk's rule Man01
• Proposition 2.2: dimension condition
• Example 2.3
• Example 3.1
• Lemma 3.2: St. Dizier--Yong StDY22
• proof
• Example 3.3