General Capelli-type identities
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911201957511168 |
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| author | Jing, Naihuan Liu, Yinlong Zhang, Jian |
| author_facet | Jing, Naihuan Liu, Yinlong Zhang, Jian |
| contents | The classical Capelli identity is an important determinantal identity of a matrix with noncommutative entries that determines the center of the enveloping algebra of the general linear Lie algebra, and was used by Weyl as a main tool to study irreducible representations in his famous book on classical groups.
In 1996 Okounkov found higher Capelli identities involving immanants of the generating matrix of $U(gl(n))$ which correspond to arbitrary orthogonal idempotent of the symmetric group. It turns out that Williamson also discovered a general Capelli identity of immanants for $U(gl(n))$ in 1981. In this paper, we use a new method to derive a family of even more general Capelli identities that include the aforementioned Capelli identities as special cases as well as many other Capelli-type identities as corollaries. In particular, we obtain generalized Turnbull's identities for both symmetric and antisymmetric matrices, as well as the generalized Howe-Umeda-Kostant-Sahi identities for antisymmetric matrices which confirm the conjecture of Caracciolo, Sokal, and Sportiello. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_14573 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | General Capelli-type identities Jing, Naihuan Liu, Yinlong Zhang, Jian Quantum Algebra Combinatorics Representation Theory Primary: 17B37 Secondary: 20G05, 17B35, 17B66, 05E10 The classical Capelli identity is an important determinantal identity of a matrix with noncommutative entries that determines the center of the enveloping algebra of the general linear Lie algebra, and was used by Weyl as a main tool to study irreducible representations in his famous book on classical groups. In 1996 Okounkov found higher Capelli identities involving immanants of the generating matrix of $U(gl(n))$ which correspond to arbitrary orthogonal idempotent of the symmetric group. It turns out that Williamson also discovered a general Capelli identity of immanants for $U(gl(n))$ in 1981. In this paper, we use a new method to derive a family of even more general Capelli identities that include the aforementioned Capelli identities as special cases as well as many other Capelli-type identities as corollaries. In particular, we obtain generalized Turnbull's identities for both symmetric and antisymmetric matrices, as well as the generalized Howe-Umeda-Kostant-Sahi identities for antisymmetric matrices which confirm the conjecture of Caracciolo, Sokal, and Sportiello. |
| title | General Capelli-type identities |
| topic | Quantum Algebra Combinatorics Representation Theory Primary: 17B37 Secondary: 20G05, 17B35, 17B66, 05E10 |
| url | https://arxiv.org/abs/2307.14573 |