B型对合的不动点Worpitzky恒等式与正二项变换

中文摘要 AI 辅助

设$\boldsymbol{\text{hyperoctahedral群}\boldsymbol{\frak B}_n}$的对合集合为$\boldsymbol{\frak I}_n^B$，$\boldsymbol{\text{相对于自然Coxeter序的下降数}}$记为$\boldsymbol{\text{des}}^B$。我们推导了不动点细化的Worpitzky恒等式：$\boldsymbol{\text{当}\boldsymbol{\frak F}_n(p,t)=\boldsymbol{\text{对所有}\boldsymbol{\frak I}_n^B\text{中的}\boldsymbol{\frak \text{π},p^{\text{fixB(π)}}t^{\text{des}}(\text{π})}\text{时，}\boldsymbol{\text{求和式}\boldsymbol{\text{Σ}}_{n≥0}\frac{\boldsymbol{\frak F}_n(p,t)\boldsymbol{z}^n}{(1-t)^{n+1}}=\boldsymbol{\text{Σ}}_{m≥0}\frac{(1+pz)^m\boldsymbol{t}^m}{(1-pz)^{m+1}(1-z^2)^{m(m+1)}}}$。提取含$j$个2-循环和$f$个不动点的层，得到Wan、Gao、Li和Yang的无不动点Worpitzky级数的单参数变形。在变量替换$\boldsymbol{x=t/(1+t)^2}$后，该变形成为正二项变换。更准确地说，若$\boldsymbol{P_j(x)=\boldsymbol{\text{Σ}}_s D_{2j,s}x^s}$为无不动点的$\boldsymbol{\text{γ-多项式}}$，则变换系数$\boldsymbol{A_{j,r}(x)}$由$\boldsymbol{\text{Σ}}_{r≥0}A_{j,r}(x)W^r=\boldsymbol{\text{Σ}}_{s=0}^j D_{2j,s}x^s (1+4xW)^{2j-2s}(1+2W+4xW^2)^s}$确定，且$(j,f)$层的$\boldsymbol{\text{γ-多项式}}$为$\boldsymbol{\text{Φ}}_{j,f}(x)=\boldsymbol{\text{Σ}}_{r=0}^f \binom{f}{r}A_{j,r}(x)$。这个明显为正的变换是本文的主要结构结果。作为推论，每个固定循环型层都是$\boldsymbol{\text{γ-正的}}$，且$\boldsymbol{\frak F}_n(p,t)$在不动点变量$p$上是系数-wise$\boldsymbol{\text{γ-正的}}$。当$f=0$和$p=1$时，分别对应Wan–Gao–Li–Yang的无不动点定理和Cao–Liu的全对合定理。我们还给出了前几个二项层及含1个和2个2-循环层的显式公式。

英文摘要

Let $\mathcal I_n^B$ be the involutions of the hyperoctahedral group $\mathfrak B_n$, and let $\des^B$ denote the descent number with respect to the natural Coxeter order. We derive the fixed-point-refined Worpitzky identity \[ \sum_{n\ge0}\frac{\mathcal F_n(p,t)\,z^n}{(1-t)^{n+1}} =\sum_{m\ge0} \frac{(1+pz)^m\,t^m}{(1-pz)^{m+1}(1-z^2)^{m(m+1)}}, \quad \mathcal F_n(p,t)=\sum_{π\in\mathcal I_n^B}p^{\fixB(π)}t^{\des^B(π)}. \] Extracting the stratum with $j$ two-cycles and $f$ fixed positions yields a one-parameter deformation of the fixed-point-free Worpitzky series of Wan, Gao, Li and Yang. After the change of variables $x=t/(1+t)^2$, this deformation becomes a positive binomial transform. More precisely, if \[ P_j(x)=\sum_sD_{2j,s}x^s \] is the fixed-point-free $γ$-polynomial, then the transform coefficients $A_{j,r}(x)$ are determined by \[ \sum_{r\ge0}A_{j,r}(x)W^r =\sum_{s=0}^{j}D_{2j,s}x^s (1+4xW)^{2j-2s}(1+2W+4xW^2)^s, \] and the $γ$-polynomial of the $(j,f)$-stratum is \[ Φ_{j,f}(x)=\sum_{r=0}^{f}\binom fr A_{j,r}(x). \] This manifestly positive transform is the main structural result of the paper. As consequences, every fixed cycle-type stratum is $γ$-positive and $\mathcal F_n(p,t)$ is coefficientwise $γ$-positive in the fixed-point variable $p$. The cases $f=0$ and $p=1$ recover, respectively, the fixed-point-free theorem of Wan--Gao--Li--Yang and the all-involution theorem of Cao--Liu. We also give explicit formulas for the first binomial layers and for the strata with one and two two-cycles.

发表机构

• College of Mathematics and Physics, Wenzhou University（温州大学数学与物理学院）
• Université Claude Bernard Lyon 1（里昂第一大学）

机构由 AI 辅助整理，请以论文原文为准。

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12 pages