  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.
