论文标题
Gelfand-Tsetlin模块：规范性和计算
Gelfand-Tsetlin modules: canonicity and calculations
论文作者
论文摘要
在本文中，我们对$ \ mathfrak {gl} _n $和图表klrw代数之间的Gelfand-Tsetlin模块之间的连接进行了更脚踏实地的介绍，并发展了一些后果。除了对较早作品中出现的Gelfand-Tsetlin模块类别的描述的新证明外，我们还显示了三个独立兴趣的新结果：（1）我们表明，每个简单的Gelfand-Tsetlin模块都是早期，Mazorchuk和Mazorchuk和Vishnyakova的典型模块，并且在两个最大的理想均具有同等的diisonical the and show nisonical of nisonical of nishical of shoonical the Onsonical nifemical nifemical nifemical nironical modions（2）可以使用Leclerc的算法进行计算双典型基础的算法来计算简单模块中的Gelfand-Tsetlin重量空间，并且（3）我们构建了$ \ Mathfrak {SL} _n $的Verma模块的基础，该模块对gelfand-tset-tsetlin subalgebebenlin subalgeberbra consists consists consists consists。 此外，我们介绍了所有等级3和4中所有积分gelfand-tsetlin模块的多重性和Gelfand-Kirillov维度的计算；不幸的是，对于排名> $> 4 $，我们的计算机不足以执行这些计算。
In this paper, we give a more down-to-earth introduction to the connection between Gelfand-Tsetlin modules over $\mathfrak{gl}_n$ and diagrammatic KLRW algebras, and develop some of its consequences. In addition to a new proof of this description of the category Gelfand-Tsetlin modules appearing in earlier work, we show three new results of independent interest: (1) we show that every simple Gelfand-Tsetlin module is a canonical module in the sense of Early, Mazorchuk and Vishnyakova, and characterize when two maximal ideals have isomorphic canonical modules, (2) we show that the dimensions of Gelfand-Tsetlin weight spaces in simple modules can be computed using an appropriate modification of Leclerc's algorithm for computing dual canonical bases, and (3) we construct a basis of the Verma modules of $\mathfrak{sl}_n$ which consists of generalized eigenvectors for the Gelfand-Tsetlin subalgebra. Furthermore, we present computations of multiplicities and Gelfand-Kirillov dimensions for all integral Gelfand-Tsetlin modules in ranks 3 and 4; unfortunately, for ranks $>4$, our computers are not adequate to perform these computations.
