Gelfand-Tsetlin Modules: Canonicity and Calculations

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.