拉斯科级数、停车函数与非交叉分拆
Lascoux series, parking functions and noncrossing partitions
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中文总结 AI 辅助
本文针对特定排列得到拉斯科多项式的组合解释，发现其与A_{n-1}型考克斯特群非交叉分拆格序复形的h多项式一致，据此提出任意有限考克斯特群h多项式的算子方法，解决了相关公开问题并得到k可除非交叉分拆h多项式的交错对称分解性质。
中文摘要 AI 辅助
在研究德马祖特征标的生成级数时，拉斯科利用等压除差分定义了由排列σ索引的多项式族$\boldsymbol{\textit{E}}_{\boldsymbol{\textit{σ}}}(\boldsymbol{\textit{t}})$，并寻求这些多项式的满意表达式。本文针对排列$\boldsymbol{\textit{σ}}=[2,3,\boldsymbol{\textit{n}},1]$或其逆，基于长度为$\boldsymbol{\textit{n}}-1$的停车函数的下降统计量，得到了$\boldsymbol{\textit{E}}_{\boldsymbol{\textit{σ}}}(\boldsymbol{\textit{t}})$的组合解释。基于拉斯科公开问题的这一进展，我们发现该特殊情形下的多项式$\boldsymbol{\textit{E}}_{\boldsymbol{\textit{σ}}}(\boldsymbol{\textit{t}})$与$\boldsymbol{A}_{\boldsymbol{n}-1}$型不可约考克斯特群$\boldsymbol{W}$对应的非交叉分拆格的序复形的$\boldsymbol{h}$多项式$\boldsymbol{h}(\boldsymbol{\textit{Δ}}(\boldsymbol{\textit{NC}}_{\boldsymbol{W}}),\boldsymbol{\textit{t}})$一致。受此巧合启发，我们为任意有限考克斯特群$\boldsymbol{W}$提出了$\boldsymbol{h}(\boldsymbol{\textit{Δ}}(\boldsymbol{\textit{NC}}_{\boldsymbol{W}}),\boldsymbol{\textit{t}})$的算子方法。作为应用，我们完全解决了阿萨纳西亚迪斯、杜夫罗普洛斯和卡拉姆波吉亚-埃万杰利努提出的关于$\boldsymbol{h}(\boldsymbol{\textit{Δ}}(\boldsymbol{\textit{NC}}_{\boldsymbol{W}}),\boldsymbol{\textit{t}})$的公开问题；对任意$\boldsymbol{k}$可除非交叉分排序集$\boldsymbol{NC}^{(\boldsymbol{k})}_{\boldsymbol{W}}$，我们还得到了其$\boldsymbol{h}$多项式$\boldsymbol{h}(\boldsymbol{\textit{Δ}}(\boldsymbol{\textit{NC}}^{(\boldsymbol{k})}_{\boldsymbol{W}}),\boldsymbol{\textit{t}})$的交错对称分解性质。
英文摘要
In the study of the generating series of Demazure characters, Lascoux used isobaric divided differences to define a family of polynomials $\mathcal{E}_σ(t)$ indexed by permutations $σ$, and asked for a satisfactory expression of these polynomials. In this paper we obtain a combinatorial interpretation of $\mathcal{E}_σ(t)$ for the permutation $σ=[2,3,\ldots,n,1]$ or its inverse in terms of the descent statistic of parking functions of length $n-1$. Based on this progress on Lascoux's open problem, we find that the polynomial $\mathcal{E}_σ(t)$ for this special case coincides with the $h$-polynomial $h(Δ(\mathrm{NC}_W),t)$ of the order complex of the noncrossing partition lattice associated to the irreducible Coxeter group $W$ of type $A_{n-1}$. We are inspired by this coincidence to give an operator approach to $h(Δ(\mathrm{NC}_W),t)$ for any finite Coxeter group $W$. As an application, we completely solve an open problem on $h(Δ(\mathrm{NC}_W),t)$ which was proposed by Athanasiadis, Douvropoulos and Kalampogia-Evangelinou. For any $k$-divisible noncrossing partition poset $\mathrm{NC}^{(k)}_W$, we also obtain the interlacing symmetric decomposition property of the $h$-polynomial $h(Δ(\mathrm{NC}^{(k)}_W),t)$.
