Formal Duality in Finite Cyclic Groups
Formal Duality in Finite Cyclic Groups
复制标题
有限循环群中的形式对偶性
DOI：
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发表时间：
2017
期刊：
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通讯作者：
R. Malikiosis
中科院分区：
文献类型：
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作者：
R. Malikiosis
有限阿贝尔群中的形式对偶概念最近出现，与欧几里得空间中的球形设计、紧球包装和能量最小化配置有关。对于有限循环群，证明了除了平凡的形式对偶对和蒂托构型之外，不存在本原形式对偶对。这一猜想已被证实为循环群的素数幂阶，以及平方自由秩序。在本文中，我们将确认猜想的其他类别的循环群，即几乎所有的循环群的阶为两个素数的幂的乘积，除了许多例外的每对素数，或其阶N满足$$pmid！mid N$$p N，其中p是满足关于N的所谓自共轭性质的素数。对于上述证明，需要各种工具：域下降法，主要用于循环阿达玛猜想，Coven和Meyerowitz的技术，通过平移平铺$$mathbb {Z}$$Z或$$mathbb {Z}_N$$ZN，这里称为多项式方法，以及分圆域的基本数论，特别是给定分圆扩张中素数的分裂。
The notion of formal duality in finite Abelian groups appeared recently in relation to spherical designs, tight sphere packings, and energy minimizing configurations in Euclidean spaces. For finite cyclic groups, it is conjectured that there are no primitive formally dual pairs besides the trivial one and the TITO configuration. This conjecture has been verified for cyclic groups of prime power order, as well as of square-free order. In this paper, we will confirm the conjecture for other classes of cyclic groups, namely almost all cyclic groups of order a product of two prime powers, with finitely many exceptions for each pair of primes, or whose order N satisfies $$pmid !mid N$$p∣∣N, where p is a prime satisfying the so-called self-conjugacy property with respect to N. For the above proofs, various tools were needed: the field descent method, used chiefly for the circulant Hadamard conjecture, the techniques of Coven and Meyerowitz for sets that tile $$mathbb {Z}$$Z or $$mathbb {Z}_N$$ZN by translations, dubbed herein as the polynomial method, as well as basic number theory of cyclotomic fields, especially the splitting of primes in a given cyclotomic extension.