We introduce the notion of essential support of a simple Gelfand-Tsetlin $\mathfrak{gl}_n$-module as an important tool towards understanding the character formula of such module. This support detects the weights in the module having maximal possible Gelfand-Tsetlin multiplicities. Using combinatorial tools we describe the essential supports of the simple socles of the universal tableaux modules. We also prove that every simple Verma module appears as a socle of a universal tableaux module and hence obtain a description of the essential supports of all simple Verma modules. As a consequence, we prove the Strong Futorny-Ovsienko Conjecture on the sharpness of the upper bounds of the Gelfand-Tsetlin multiplicities. In addition we give a very explicit description of the support and essential support of the simple singular Verma module $M(-\rho)$

中文翻译：

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Gelfand-Tsetlin 多重性的界限和 Verma 模块的画面实现

我们介绍了一个简单的 Gelfand-Tsetlin $\mathfrak{gl}_n$-module 的基本支持的概念，作为理解此类模块的字符公式的重要工具。这种支持检测模块中具有最大可能 Gelfand-Tsetlin 多重性的权重。使用组合工具，我们描述了通用画面模块的简单社会的基本支持。我们还证明了每个简单的 Verma 模块都作为一个通用的 tableaux 模块出现，因此获得了所有简单 Verma 模块的基本支持的描述。因此，我们证明了关于 Gelfand-Tsetlin 多重性上限锐度的强 Futorny-Ovsienko 猜想。此外，我们非常明确地描述了简单单数 Verma 模块 $M(-\rho)$ 的支持和基本支持