Stretched Schubert coefficients are eventually quasi-polynomial

TL;DR

The paper proves that stretched Schubert coefficients are eventually quasi-polynomial, confirming Kirillov's conjecture and using combinatorics and Ehrhart theory to analyze their properties.

Contribution

It establishes the eventual quasi-polynomial nature of stretched Schubert coefficients, proving a longstanding conjecture and providing new insights into their generating functions.

Findings

Proves that $f_{u,v,w}(N)$ is eventually quasi-polynomial.

Shows the generating function for $f_{u,v,w}(N)$ is rational.

Provides counterexamples to the saturation conjecture for 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.