发表机构
University of Kansas（堪萨斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文否证 Reiner-Shimozono 猜想，通过放宽 Demazure 模关系得到精确系数提取公式，构造无穷反例族，并给出多项式时间可检验的原子系数正性充分判据。
AI 中文摘要
我们否证了 Reiner--Shimozono 猜想，该猜想声称 key 多项式的乘积在 Demazure 原子中具有非负展开。通过放宽 Demazure 模的定义关系，我们获得了一个精确的系数提取公式，该公式产生了一个显式的无穷反例族，其中包括在二十八个变量中的负系数。这些例子还否证了 Polo 关于 Schubert 簇上截面模张量积的 Schubert 滤过的猜想，即使允许连续商由 Schubert 簇并集上的截面空间给出。对于单个原子系数，我们给出了一个正性的充分判据，该判据可在二进制输入长度的多项式时间内检验，且对秩一致。对于每个固定秩 $n\ge4$，当组合项条目的上界趋于无穷时，该判据能识别出非零比例的正原子系数。
英文摘要
We disprove the Reiner--Shimozono conjecture that products of key polynomials have nonnegative expansions in Demazure atoms. By relaxing the defining relations of Demazure modules, we obtain an exact coefficient-extraction formula that yields an explicit infinite family of counterexamples, including negative coefficients in twenty-eight variables. These examples answer van der Kallen's long-standing open question on relative Schubert filtrations negatively, already for tensor products of two dual Joseph modules. For individual atom coefficients, we give a sufficient criterion for positivity that is checkable in polynomial time in the binary input length, uniformly in rank. For each fixed rank $n\ge4$, this criterion recognizes a nonvanishing proportion of the positive atom coefficients as the bound on the composition entries tends to infinity.
Comments23 pages; corrected and expanded the historical discussion of Polo's Schubert-filtration conjecture, van der Kallen's question, and the Reiner--Shimozono conjecture. Clarified the exposition in several places
