Immanant inequalities and weight spaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911127440457728 |
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| author | Jing, Naihuan Liu, Yinlong Zhang, Jian |
| author_facet | Jing, Naihuan Liu, Yinlong Zhang, Jian |
| contents | We first obtain a trace formula for immanants of generalized principal submatrix of any complex matrix based on any weight space for finite dimensional representations of the general linear group. Our trace formula contains Kostant's famous formula for immanants on $0$-weight spaces as special case. We then present a criterion for non-vanishing immanants for any generalized principal submatrix of positive definite Hermitian or nonsingular totally nonnegative matrices, which strengthened the well-known results of Schur and Stembridge. Furthermore, we present an inequality that contains Kostant, Schur and Stembridge's famous inequalities as special cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_20382 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Immanant inequalities and weight spaces Jing, Naihuan Liu, Yinlong Zhang, Jian Representation Theory Primary: 20G05 Secondary: 15A15, 17B35, 05E10 We first obtain a trace formula for immanants of generalized principal submatrix of any complex matrix based on any weight space for finite dimensional representations of the general linear group. Our trace formula contains Kostant's famous formula for immanants on $0$-weight spaces as special case. We then present a criterion for non-vanishing immanants for any generalized principal submatrix of positive definite Hermitian or nonsingular totally nonnegative matrices, which strengthened the well-known results of Schur and Stembridge. Furthermore, we present an inequality that contains Kostant, Schur and Stembridge's famous inequalities as special cases. |
| title | Immanant inequalities and weight spaces |
| topic | Representation Theory Primary: 20G05 Secondary: 15A15, 17B35, 05E10 |
| url | https://arxiv.org/abs/2508.20382 |