发表机构
School of Mathematics and Statistics, Beijing Jiaotong University（北京交通大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对配对矩阵引入部分扭曲多项式框架，证明存在满足特定关系的局部运算，并建立递推关系以计算相关多项式。
AI 中文摘要
Gross、Mansour 和 Tucker [European Journal of Combinatorics, 95 (2021): 103329] 引入了带状图的部分扭曲多项式。最近，Deng、Jin 和 Yan 将部分扭曲多项式推广到矩阵代数的框架中，并研究了它们的一些基本性质。他们提出疑问：是否存在称为部分对偶 δ 和部分 Petrie 对偶 τ 的矩阵运算，作用于对 (M,A)，其中 M 是一个方阵，其行和列由有限集合 V 索引，且 A⊆V，使得 δ²=τ²=(δτ)³=id，并且部分扭曲多项式的指数与通过对 (M,A) 应用 ● 所得到的矩阵的某个参数一致。在本文中，我们引入了二元域 GF(2) 上部分扭曲多项式的配对矩阵框架。我们证明了在 ((M,I_{|V|}),A) 上存在两个局部运算 δ 和 τ，满足 δ²=τ²=(δτ)³=id 且对于 ●∈{δ,τ,δτ,τδ,δτδ}，有 P_{⟨●⟩}((M,I_{|V|}),z)=P_{⟨●⟩}(M,z)，从而肯定地回答了他们的疑问。最后，我们建立了关于一条边的部分 ⟨δτδ⟩-多项式的递推关系。该递推关系使得能够计算某些花束、简单图和简单符号图的部分 ⟨δτδ⟩-多项式。
英文摘要
Gross, Mansour, and Tucker~[European Journal of Combinatorics, 95 (2021): 103329] introduced the \emph{partial-twuality polynomials} of ribbon graphs. Recently, Deng, Jin, and Yan generalized the partial-twuality polynomials to the framework of matrix algebra and investigated several of their basic properties. They asked whether there exist matrix operations, called partial duality $δ$ and partial Petrie duality $τ$, on pairs $(M,A)$, where $M$ is a square matrix whose rows and columns are indexed by a finite set $V$ and $A\subseteq V$, such that $δ^2=τ^2=(δτ)^3=id$ and the exponent of the partial-twuality polynomials coincides with some parameter of the matrix obtained by applying \(\bullet\) to \((M, A)\). In this paper, we introduce a paired-matrix framework for partial-twuality polynomials over the binary field $\mathbb{GF}(2)$. We prove that there exist two local operations \(δ\) and \(τ\) on \(\bigl((M,I_{|V|}),A\bigr)\) satisfying $δ^2=τ^2=(δτ)^3=id$ and $P_{\langle \bullet \rangle}((M,I_{|V|}),z)=P_{\langle \bullet \rangle}(M,z)$ for $\bullet \in \{δ, τ, δτ, τδ, δτδ\}$, thereby answering their question affirmatively. Finally, we establish a recurrence relation for the partial \(\langleδτδ\rangle\)-polynomial with respect to an edge. This recurrence enables the computation of the partial \(\langleδτδ\rangle\)-polynomial for certain bouquets, simple graphs, and simple signed graphs.
Comments18 pages
