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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.15680 |
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| _version_ | 1866916665281740800 |
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| author | González-Serrano, Luis Angel Maximenko, Egor A. |
| author_facet | González-Serrano, Luis Angel Maximenko, Egor A. |
| contents | We consider polynomials of the form $\operatorname{s}_λ(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $λ$ is an integer partition, $\operatorname{s}_λ$ is the Schur polynomial associated to $λ$, and $y_j^{[\varkappa_j]}$ denotes $y_j$ repeated $\varkappa_j$ times. We represent $\operatorname{s}_λ(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$ as a quotient whose the denominator is the determinant of the confluent Vandermonde matrix, and the numerator is the determinant of some generalized confluent Vandermonde matrix. We give three algebraic proofs of this formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_15680 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bialternant formula for Schur polynomials with repeating variables González-Serrano, Luis Angel Maximenko, Egor A. Combinatorics Numerical Analysis 05E05, 15A15 We consider polynomials of the form $\operatorname{s}_λ(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $λ$ is an integer partition, $\operatorname{s}_λ$ is the Schur polynomial associated to $λ$, and $y_j^{[\varkappa_j]}$ denotes $y_j$ repeated $\varkappa_j$ times. We represent $\operatorname{s}_λ(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$ as a quotient whose the denominator is the determinant of the confluent Vandermonde matrix, and the numerator is the determinant of some generalized confluent Vandermonde matrix. We give three algebraic proofs of this formula. |
| title | Bialternant formula for Schur polynomials with repeating variables |
| topic | Combinatorics Numerical Analysis 05E05, 15A15 |
| url | https://arxiv.org/abs/2312.15680 |