Gelfand-Tsetlin modules: canonicity and calculations
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| Format: | Preprint |
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2020
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| author | Silverthorne, Turner Webster, Ben |
| author_facet | Silverthorne, Turner Webster, Ben |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_06029 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Gelfand-Tsetlin modules: canonicity and calculations Silverthorne, Turner Webster, Ben Representation Theory Rings and Algebras 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. |
| title | Gelfand-Tsetlin modules: canonicity and calculations |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2011.06029 |