Constructions of primitive formally dual pairs having subsets with unequal sizes
Constructions of primitive formally dual pairs having subsets with unequal sizes
复制标题
具有大小不等子集的原始形式对偶对的构造
DOI：
--
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
A. Pott
中科院分区：
文献类型：
--
作者：
Shuxing Li;A. Pott
形式对偶的概念是由Cohn，Kumar和Schürmann提出的，它反映了能量最小化周期性构型之间的显著对称性。这种形式上的对偶性后来被科恩、库马尔、赖赫和舒尔曼转化为纯粹的组合性质，其中相应的组合对象被称为形式上的对偶对。受能量最小化问题上这一令人惊讶的应用的启发，我们重点关注原始形式对偶对的代数构造。值得注意的是，几乎所有已知的原始形式对偶对的例子都满足两个子集具有相同的大小。事实上，在这项工作之前，只有一个已知的例子来自计算机搜索，它在Z 2 × Z 4 2中具有大小不等的子集。受这个例子的启发，我们提出了一个提升构造框架和一个递归构造框架，它们从已知的形式对偶对生成新的原语形式对偶对。作为应用，当m ≥ 2时，我们得到了Z2 × Z42m中m + 1个两两不等的本原形式对偶偶，它们的子集大小不等.
The concept of formal duality was proposed by Cohn, Kumar and Schürmann, which reflects a remarkable symmetry among energy‐minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Schürmann, where the corresponding combinatorial objects were called formally dual pairs. Motivated by this surprising application on the energy minimization problem, we focus on the algebraic constructions of primitive formally dual pairs. It is worth noting that almost all known examples of primitive formally dual pairs satisfy that the two subsets have the same size. Indeed, prior to this work, there was only one known example derived from computer search, which had subsets with unequal sizes in Z 2 × Z 4 2 . Inspired by this example, we propose a lifting construction framework and a recursive construction framework, which generate new primitive formally dual pairs from known ones. As an application, for m ≥ 2 , we obtain m + 1 pairwise inequivalent primitive formally dual pairs in Z 2 × Z 4 2 m , which have subsets with unequal sizes.
有限阿贝尔群中的形式对偶性
DOI：
10.1016/j.jcta.2018.11.005
发表时间：
2019
期刊：
J. Comb. Theory, Ser. A
影响因子：
--
作者：
Shuxing Li;Alexander Pott;Robert Schüler
通讯作者：
Robert Schüler