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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2501.04605 |
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| _version_ | 1866908884737720320 |
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| author | Ngo, Nhok Tkhai Shon |
| author_facet | Ngo, Nhok Tkhai Shon |
| contents | In this paper, we consider the big algebra recently introduced by Hausel for the $\mathrm{GL}_n$-action on the coordinate ring of the matrix space $\mathrm{Mat}(n,r)$. In particular, we obtain explicit formulas for the big algebra generators in terms of differential operators with polynomial coefficients. We show that big algebras in type $A$ are commutative and relate them to the Bethe subalgebra in the Yangian $\operatorname{Y}(\mathfrak{gl}_{n})$. We apply these results to big algebras of symmetric powers of the standard representation of $\mathrm{GL}_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04605 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Big Algebra in Type $A$ for the Coordinate Ring of the Matrix Space Ngo, Nhok Tkhai Shon Representation Theory Quantum Algebra In this paper, we consider the big algebra recently introduced by Hausel for the $\mathrm{GL}_n$-action on the coordinate ring of the matrix space $\mathrm{Mat}(n,r)$. In particular, we obtain explicit formulas for the big algebra generators in terms of differential operators with polynomial coefficients. We show that big algebras in type $A$ are commutative and relate them to the Bethe subalgebra in the Yangian $\operatorname{Y}(\mathfrak{gl}_{n})$. We apply these results to big algebras of symmetric powers of the standard representation of $\mathrm{GL}_n$. |
| title | Big Algebra in Type $A$ for the Coordinate Ring of the Matrix Space |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2501.04605 |