标题： 抛物线 Verma 模块特征的对数凹度以及受限 Kostant 配分函数 显示英文标题

标题： Log-concavity of characters of parabolic Verma modules, and of restricted Kostant partition functions

摘要： 2022年，Huh-Matherne-Meszaros-St. Dizier证明了归一化的施尔多项式是洛伦兹的，从而在正象限上（分别在它们的支撑集上，类型-$A$的根方向上）得出它们的连续（分别离散）对数凹性。这一结果的重新解释是，$\mathfrak{sl}_{n+1}(\mathbb{C})$的有限维简单表示的特征是未归一化的洛伦兹。在同一篇论文中，这些作者还证明了$\mathfrak{sl}_{n+1}(\mathbb{C})$上的韦尔模的偏移特征是未归一化的洛伦兹。在这项工作中，我们将这些结果扩展到一个更大的模块族，该族包含了上述两者：我们证明了$\mathfrak{sl}_{n+1}(\mathbb{C})$上所有抛物线韦尔模的偏移特征是未归一化的洛伦兹。证明涉及$[n+1]$上的某些图；更强烈地，我们解释了为什么任何无环多重图在$[n+1]$上的科斯滕分拆函数的特征（即生成函数）在经过平移和归一化后是洛伦兹的。相比之下，我们证明了一个更大的最高权模的通用族，即高阶韦尔模，其特征并不具有离散对数凹性。最后，我们将所有这些结果扩展到半单李代数$\oplus_{t=1}^T \mathfrak{sl}_{n_t+1}(\mathbb{C})$上的抛物线（即“一阶”）和高阶韦尔模。 显示英文摘要

摘要： In 2022, Huh-Matherne-Meszaros-St. Dizier showed that normalized Schur polynomials are Lorentzian, thereby yielding their continuous (resp. discrete) log-concavity on the positive orthant (resp. on their support, in type-$A$ root directions). A reinterpretation of this result is that the characters of finite-dimensional simple representations of $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian. In the same paper, these authors also showed that shifted characters of Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian. In this work we extend these results to a larger family of modules that subsumes both of the above: we show that shifted characters of all parabolic Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian. The proof involves certain graphs on $[n+1]$; more strongly, we explain why the character (i.e., generating function) of the Kostant partition function of any loopless multigraph on $[n+1]$ is Lorentzian after shifting and normalizing. In contrast, we show that a larger universal family of highest weight modules, the higher order Verma modules, do not have discretely log-concave characters. Finally, we extend all of these results to parabolic (i.e. "first order") and higher order Verma modules over the semisimple Lie algebras $\oplus_{t=1}^T \mathfrak{sl}_{n_t+1}(\mathbb{C})$.