模理想化子链与具有重复部分的划分的不可细分性
A modular idealizer chain and unrefinability of partitions with repeated parts
AI总结：
本文针对p>2时对称群正规化子链计算的困难，提出在Z_m李环上以整数划分表示元素的模理想化子链框架，证明其由至多m-1部分的划分生成，并在m=2时建立与正规化子链的对应关系。
摘要：
最近Aragona等人在Sym(2^n)的一个Sylow 2-子群中引入了一条正规化子链，从一个初等阿贝尔正则子群开始。他们证明了链中连续群的指数取决于划分为不同部分的划分数量，并通过刚性换位子描述了链中的前n-2项。此外，他们证明了链的第(n-1)项通过对应于划分为不同部分的不可细分划分的刚性换位子来描述。尽管上述链可以在Sym(p^n)的Sylow p-子群中定义，但对于p > 2，在缺乏刚性换位子适当概念的情况下，计算正规化子链成为一项具有挑战性的任务。本文从另一个角度解决了这个问题。我们为正规化子链提出了一个更一般的框架，在Z_m上的一个李环中定义了一条理想化子链，其元素由整数划分表示。我们展示了相应的理想化子如何由至多m-1部分的划分子集生成，并猜想理想化子链的增长方式与对称群中的正规化子链相同。作为证据，我们在m=2的情况下建立了两种构造之间的对应关系。
英文摘要：
Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2^n), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n-2 terms in the chain. Moreover, they proved that the (n-1)-th term of the chain is described by means of rigid commutators corresponding to unrefinable partitions into distinct parts. Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(p^n), for p > 2 computing the chain of normalizers becomes a challenging task, in the absence of a suitable notion of rigid commutators. This problem is addressed here from an alternative point of view. We propose a more general framework for the normalizer chain, defining a chain of idealizers in a Lie ring over Z_m whose elements are represented by integer partitions. We show how the corresponding idealizers are generated by subsets of partitions into at most m-1 parts and we conjecture that the idealizer chain grows as the normalizer chain in the symmetric group. As an evidence of this, we establish a correspondence between the two constructions in the case m=2.
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