The monomial expansions of modified Macdonald polynomials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912467967279104 |
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| author | Jin, Emma Yu Lin, Xiaowei |
| author_facet | Jin, Emma Yu Lin, Xiaowei |
| contents | We discover a family $A$ of sixteen statistics on fillings of any given Young diagram and prove new combinatorial formulas for modified Macdonald polynomials, that is, $$\tilde{H}_λ(X;q,t)=\sum_{σ\in T(λ)}x^σq^{maj(σ)}t^{η(σ)}$$ for each statistic $η\in A$. Building upon this new formula, we establish four compact formulas for the modified Macdonald polynomials, namely, $$\tilde{H}_λ(X;q,t)=\sum_σd_{\varepsilon}(σ)x^σq^{maj(σ)}t^{η(σ)}$$ which is summed over all canonical or dual canonical fillings of a Young diagram and $d_{\varepsilon}(σ)$ is a product of $t$-multinomials. Finally, the compact formulas enable us to derive four explicit expressions for the monomial expansion of modified Macdonald polynomials, one of which coincides with the formula given by Garbali and Wheeler (2020). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23373 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The monomial expansions of modified Macdonald polynomials Jin, Emma Yu Lin, Xiaowei Combinatorics Mathematical Physics 05E05, 33D52, 05A19 We discover a family $A$ of sixteen statistics on fillings of any given Young diagram and prove new combinatorial formulas for modified Macdonald polynomials, that is, $$\tilde{H}_λ(X;q,t)=\sum_{σ\in T(λ)}x^σq^{maj(σ)}t^{η(σ)}$$ for each statistic $η\in A$. Building upon this new formula, we establish four compact formulas for the modified Macdonald polynomials, namely, $$\tilde{H}_λ(X;q,t)=\sum_σd_{\varepsilon}(σ)x^σq^{maj(σ)}t^{η(σ)}$$ which is summed over all canonical or dual canonical fillings of a Young diagram and $d_{\varepsilon}(σ)$ is a product of $t$-multinomials. Finally, the compact formulas enable us to derive four explicit expressions for the monomial expansion of modified Macdonald polynomials, one of which coincides with the formula given by Garbali and Wheeler (2020). |
| title | The monomial expansions of modified Macdonald polynomials |
| topic | Combinatorics Mathematical Physics 05E05, 33D52, 05A19 |
| url | https://arxiv.org/abs/2506.23373 |