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Gelfand-Tsetlin Modules: Canonicity and Calculations

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Abstract

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.

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All code used in the production of data in this file is available at: https://github.com/bwebste/gelfand-tsetlin-public

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Acknowledgements

Many thanks to Jon Brundan for generously sharing computer code that our program was based on, and to Pablo Zadunaisky, Elizaveta Vishnyakova, Walter Mazorchuk and Jonas Hartwig for helpful discussions about their work.

Funding

T. S. was supported by NSERC and the University of Waterloo through an Undergraduate Student Research Award. B. W. is supported by an NSERC Discovery Grant. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, Canada

    Turner Silverthorne

  2. Department of Pure Mathematics, University of Waterloo and Perimeter Institute for Mathematical Physics, Waterloo, ON, Canada

    Ben Webster

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  1. Turner Silverthorne
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Correspondence to Ben Webster.

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Presented by: Michela Varagnolo

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Silverthorne, T., Webster, B. Gelfand-Tsetlin Modules: Canonicity and Calculations. Algebr Represent Theor (2024). https://doi.org/10.1007/s10468-024-10264-y

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