Modified Macdonald polynomials and mu-Mahonian statistics
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909797996036096 |
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| author | Jin, Emma Yu Lin, Xiaowei |
| author_facet | Jin, Emma Yu Lin, Xiaowei |
| contents | The Haglund--Haiman--Loehr theorem provides the following combinatorial formula for the modified Macdonald polynomials: $$\tilde{H}_μ(X;q,t)=\sum_{σ: μ\rightarrow \mathbb{P}}x^σt^{maj(σ)}q^{inv(σ)}.$$ Inspired by Martin's multiline-queue formula for the stationary distribution of multitype asymmetric simple exclusion processes, Corteel, Haglund, Mandelshtam, Mason and Williams recently introduced the queue inversion statistic $quinv$ and conjectured that the tableaux formula for $\tilde{H}_μ(X;q,t)$ is invariant if the inversion statistic $inv$ is replaced by $quinv$. This was subsequently resolved by Ayyer, Mandelshtam and Martin, who proposed a stronger conjecture on the equivalence of the two refined formulas for $\tilde{H}_μ(X;q,t)$.
Our main result confirms this Ayyer--Mandelshtam--Martin conjecture. We establish an equidistribution between the pairs $(inv,maj)$ and $(quinv,maj)$ of $μ$-Mahonian statistics on any row-equivalency class $[τ]$, where $τ$ is a filling of the Young diagram of $μ$. As a byproduct of our approach, we show that if $τ$ is a rectangular filling, the triples $(inv,quinv,maj)$ and $(quinv,inv,maj)$ have the same distribution over $[τ]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14099 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modified Macdonald polynomials and mu-Mahonian statistics Jin, Emma Yu Lin, Xiaowei Combinatorics 05E05, 05A19 The Haglund--Haiman--Loehr theorem provides the following combinatorial formula for the modified Macdonald polynomials: $$\tilde{H}_μ(X;q,t)=\sum_{σ: μ\rightarrow \mathbb{P}}x^σt^{maj(σ)}q^{inv(σ)}.$$ Inspired by Martin's multiline-queue formula for the stationary distribution of multitype asymmetric simple exclusion processes, Corteel, Haglund, Mandelshtam, Mason and Williams recently introduced the queue inversion statistic $quinv$ and conjectured that the tableaux formula for $\tilde{H}_μ(X;q,t)$ is invariant if the inversion statistic $inv$ is replaced by $quinv$. This was subsequently resolved by Ayyer, Mandelshtam and Martin, who proposed a stronger conjecture on the equivalence of the two refined formulas for $\tilde{H}_μ(X;q,t)$. Our main result confirms this Ayyer--Mandelshtam--Martin conjecture. We establish an equidistribution between the pairs $(inv,maj)$ and $(quinv,maj)$ of $μ$-Mahonian statistics on any row-equivalency class $[τ]$, where $τ$ is a filling of the Young diagram of $μ$. As a byproduct of our approach, we show that if $τ$ is a rectangular filling, the triples $(inv,quinv,maj)$ and $(quinv,inv,maj)$ have the same distribution over $[τ]$. |
| title | Modified Macdonald polynomials and mu-Mahonian statistics |
| topic | Combinatorics 05E05, 05A19 |
| url | https://arxiv.org/abs/2407.14099 |