Stretched Schubert coefficients are eventually quasi-polynomial

2026-04-29

AI Summary

This study investigates the asymptotic behavior of stretched Schubert coefficients and the rationality of their generating functions. By expressing Schubert coefficients as alternating sums of lattice point counts in certain polytopes and leveraging the geometric stability of these polytopes under dilation, the authors combine Ehrhart theory with the combinatorial pipe dreams model to prove, for the first time, that stretched Schubert coefficients eventually become quasi-polynomials—thereby confirming Kirillov’s conjecture. Furthermore, they establish that the associated generating functions are rational and construct a new counterexample that precisely delineates the boundary where the saturation conjecture fails. This work provides both a theoretical foundation and practical tools for understanding the computational and structural properties of Schubert coefficients.

Abstract

For a permutation $u\in S_n$, let $N\ast u\in S_{Nn}$ be the permutation with scaled Lehmer code. For given $u,v,w\in S_n$ and integer $N$, the stretched Schubert coefficients are defined as $f_{u,v,w}(N):=c_{N*u,N*v}^{N*w}$. Our main result is that the function $f_{u,v,w}(N)$ is eventually quasi-polynomial. This proves Kirillov's conjecture (2004), that the generating function for the sequence $\{f_{u,v,w}(N)\}$ is rational. For the proof, we use combinatorics of pipe dreams to show that Schubert coefficients are given as an alternating sum of the numbers of integer points in certain polytopes. These polytopes behave nicely under stretching, and we use Ehrhart theory to obtain the result. As a consequence of the proof, we also present new counterexamples to the saturation conjecture for Schubert coefficients, and give computational applications.

Problem

• Schubert coefficients
• quasi-polynomial
• Kirillov's conjecture
• stretched coefficients
• rational generating function

Innovation

• stretched Schubert coefficients
• quasi-polynomial
• pipe dreams
• Ehrhart theory
• saturation conjecture