Hadamard Products of dual Jacobi-Trudi matrices
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911261454761984 |
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| author | Angarone, Robert Kim, Jang Soo Oh, Jaeseong Soskin, Daniel |
| author_facet | Angarone, Robert Kim, Jang Soo Oh, Jaeseong Soskin, Daniel |
| contents | We study positivity properties of Hadamard products of Jacobi-Trudi matrices. Maló proved that the Hadamard (entrywise) product of two totally positive upper-triangular Toeplitz matrices whose Toeplitz sequences are the coefficient sequences of real-rooted polynomials with nonpositive zeros is again totally positive. Sokal conjectured that this result can be strengthened to total monomial positivity for the Hadamard product of Jacobi-Trudi matrices. In this paper we show that Temperley-Lieb immanants are Schur positive for Hadamard products of Jacobi-Trudi matrices given by ribbon-like skew shapes. In particular, we affirm Sokal's conjecture for minors given by ribbon-like skew shapes. Moreover, we provide a manifestly positive Schur expansion for Temperley-Lieb immanants evaluated on the Hadamard product of Jacobi-Trudi matrices indexed by ribbons. In addition, for the ribbon case, we construct a corresponding representation, offering a representation-theoretic proof of the Schur positivity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08969 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hadamard Products of dual Jacobi-Trudi matrices Angarone, Robert Kim, Jang Soo Oh, Jaeseong Soskin, Daniel Combinatorics 05E05, 05A05, 05E10, 15B05, 15B48 We study positivity properties of Hadamard products of Jacobi-Trudi matrices. Maló proved that the Hadamard (entrywise) product of two totally positive upper-triangular Toeplitz matrices whose Toeplitz sequences are the coefficient sequences of real-rooted polynomials with nonpositive zeros is again totally positive. Sokal conjectured that this result can be strengthened to total monomial positivity for the Hadamard product of Jacobi-Trudi matrices. In this paper we show that Temperley-Lieb immanants are Schur positive for Hadamard products of Jacobi-Trudi matrices given by ribbon-like skew shapes. In particular, we affirm Sokal's conjecture for minors given by ribbon-like skew shapes. Moreover, we provide a manifestly positive Schur expansion for Temperley-Lieb immanants evaluated on the Hadamard product of Jacobi-Trudi matrices indexed by ribbons. In addition, for the ribbon case, we construct a corresponding representation, offering a representation-theoretic proof of the Schur positivity. |
| title | Hadamard Products of dual Jacobi-Trudi matrices |
| topic | Combinatorics 05E05, 05A05, 05E10, 15B05, 15B48 |
| url | https://arxiv.org/abs/2511.08969 |