关于Schubert多项式滑动分解的若干结果
Some results related to the slide decomposition of Schubert polynomials
AI总结：
该研究推广子词复形相关证明方法，验证Schubert多项式的森林多项式展开对应子词复形子球求和，给出Monk法则等的双射与表复形相关结论，否定了滑动多项式是矩阵Schubert簇Gröbner退化的猜想。
摘要：
Schubert多项式展开为滑动多项式的过程对应于子词复形中子球的求和。近期学界对Schubert多项式的其他更粗粒度展开产生了研究兴趣。我们将文献[KM04]中用于证明子词复形是球或球面的方法推广为更通用的方法，并利用该方法证明：Schubert多项式展开为森林多项式的过程同样对应于子词复形中子球的求和。\n将两个Schubert多项式的乘积$\u27d6_π\u27d6_ρ$展开为Schubert多项式$\u27d6_σ$时，π与ρ的既约字的洗牌和σ的既约字（带重数计数）之间存在双射。我们针对Monk法则和Sottile的Pieri法则给出了这样的双射。\n我们给出了滑动多项式、glide多项式和基本拟对称多项式的基于表的定义，并证明Schur多项式展开为基本拟对称多项式的过程对应于表复形中子球的求和。\nSchubert多项式是矩阵Schubert簇的上同调类，但目前尚无滑动多项式的几何解释。我们证明了滑动多项式不是矩阵Schubert簇的（反对角）Gröbner退化，从而否定回答了文献[ST21, 第1.4节]中的一个问题。
英文摘要：
The expansion of a Schubert polynomial into slide polynomials corresponds to a sum over sub-balls in the subword complex. There has been recent interest in other, coarser, expansions of Schubert polynomials. We extend the methods used in [KM04] to prove that the subword complex is a ball or a sphere to a more general method, and use it to prove that the expansion of a Schubert polynomial into forest polynomials also corresponds to a sum over sub-balls in the subword complex. When expanding the product $\mathfrak{S}_π\mathfrak{S}_ρ$ of two Schubert polynomials into Schubert polynomials $\mathfrak{S}_σ$, there is a bijection between shuffles of reduced words for $π$ and $ρ$ and reduced words for $σ$ (counted with multiplicity). We give such a bijection for Monk's rule and Sottile's Pieri rule. We give tableau-based definitions of slide polynomials, glide polynomials, and fundamental quasisymmetric polynomials and show that the expansion of a Schur polynomial into fundamental quasisymmetric polynomials corresponds to a sum over sub-balls in the tableau complex. The Schubert polynomials are the cohomology classes of matrix Schubert varieties, but there is no geometric explanation of the slide polynomials. We show that the slide polynomials are not (antidiagonal) Gröbner degenerations of matrix Schubert varieties, answering in the negative a question of [ST21, Section 1.4].
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