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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.04101 |
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| _version_ | 1866910356079640576 |
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| author | Ross, Julius Wu, Kuang-Yu |
| author_facet | Ross, Julius Wu, Kuang-Yu |
| contents | To any Schur polynomial $s_λ$ one can associated its derived polynomials $s_λ{(i)}$ $i=0,\ldots,|λ|$ by the rule $$s_λ(x_1+t,\ldots,x_n+t) = \sum_i s_λ^{(i)}(x_1,\ldots,x_n) t^i.$$ We conjecture that $$(s_λ^{(i)})^2 - s_λ^{(i-1)} s_λ^{(i+1)}$$ is always Schur positive and prove this when $i=1$ for rectangles $λ= (k^\ell)$, for hooks $λ= (k, 1^{\ell -1})$, and when $λ= (k,k,1)$ or $λ= (3,2^{k-1})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04101 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Schur positivity of difference of products of derived Schur polynomials Ross, Julius Wu, Kuang-Yu Combinatorics 05E05 To any Schur polynomial $s_λ$ one can associated its derived polynomials $s_λ{(i)}$ $i=0,\ldots,|λ|$ by the rule $$s_λ(x_1+t,\ldots,x_n+t) = \sum_i s_λ^{(i)}(x_1,\ldots,x_n) t^i.$$ We conjecture that $$(s_λ^{(i)})^2 - s_λ^{(i-1)} s_λ^{(i+1)}$$ is always Schur positive and prove this when $i=1$ for rectangles $λ= (k^\ell)$, for hooks $λ= (k, 1^{\ell -1})$, and when $λ= (k,k,1)$ or $λ= (3,2^{k-1})$. |
| title | Schur positivity of difference of products of derived Schur polynomials |
| topic | Combinatorics 05E05 |
| url | https://arxiv.org/abs/2403.04101 |