Gamma 向量作为反切比雪夫展开、A 型到 B 型变换及其与代数结构的联系
Gamma vectors as inverted Chebyshev expansions, type A to B transformations, and connections to algebraic structures
AI总结：
本文证明互反/回文多项式的 gamma 向量本质上是反切比雪夫多项式基展开，揭示了其与单纯复形几何组合学的联系，并将其提升至 $cd$-indices 的修改与下降统计量的计算。
摘要：
给定一个偶数次的互反/回文多项式，我们证明了 gamma 向量本质上由反切比雪夫多项式基展开给出。作为一个直接推论，我们用由一个系数固定、其余系数除以 2 的反缩放元组所构建的互反多项式的实根性，刻画了切比雪夫多项式线性组合的实根性。这可以被视为 Bel-Afia--Meroni--Telen 关于切比雪夫曲线关于原点的双曲性的一个近期结果在任意维度的对应。一般而言，在稀疏多项式求根中，Chebyshev varieties 作为 toric varieties 的对应物。除此之外，反切比雪夫展开还揭示了 gamma 向量构造的内在性质与单纯复形及偏序集的几何组合学之间的联系。\n 我们通过应用 Hetyei 关于 Tchebyshev subdivisions 和 Tchebyshev posets 的工作发现了这一点。特别地，我们发现 gamma 向量变换与通过逐次边细分将 type A Coxeter complex 变换为 type B Coxeter complex 所得到的单纯复形的 $f$-vectors 密切相关。将其提升至对 $cd$-indices 的修改，我们证明了 gamma 向量的反切比雪夫多项式展开提升为 cross polytopes（的细分）之和，该和可使用（拓扑）下降统计量计算。尽管存在许多涉及下降统计量的 gamma positivity 示例，但值得注意的是，我们仅假设输入多项式是互反/回文的。最后，由导数得到的第二类切比雪夫多项式给出了与 Hopf algebras 和拟对称函数的联系，以及由 $\mathfrak{sl}_2(\mathbb{C})$-representations 诱导的 Lefschetz-type maps。
英文摘要：
Given a reciprocal/palindromic polynomial of even degree, we show that the gamma vector is essentially given by an inverted Chebyshev polynomial basis expansion. As an immediate consequence, we characterize real-rootedness of a linear combination of Chebyshev polynomials in terms of real-rootedness of that of the reciprocal polynomial built out of an inverted scaled tuple of the coefficients with one fixed and the rest divided by 2. It can be taken as a counterpart for arbitrary dimensions of a recent result of Bel-Afia--Meroni--Telen on hyperbolicity of Chebyshev curves with respect to the origin. In general, Chebyshev varieties serve as a counterpart of toric varieties in sparse polynomial root finding. Apart from this, the inverted Chebyshev expansion also yields connections between intrinsic properties of the gamma vector construction and the geometric combinatorics of simplicial complexes and posets. We find this by applying work of Hetyei on Tchebyshev subdivisions and Tchebyshev posets. In particular, we find that the gamma vector transformation is closely related to $f$-vectors of simplicial complexes resulting from successive edge subdivisions that transform the type A Coxeter complex to the type B Coxeter complex. Lifting to this to a modification of $cd$-indices, we show that the gamma vector inverted Chebyshev polynomial expansion lifts to a sum of (subdivisions of) cross polytopes which can be computed using (topological) descent statistics. While there are many examples where gamma positivity involving descent statistics, it is interesting to note we only assume the input polynomial is reciprocal/palindromic. Finally, Chebyshev polynomials of the second kind from derivatives give connections to Hopf algebras and quasisymmetric functions along with Lefschetz-type maps induced by $\mathfrak{sl}_2(\mathbb{C})$-representations.
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