酉表示下具有不可约不忠实子集的群

arXiv: Group Theory Pub Date : 2018-07-13 DOI:10.5802/CML.61
P. Caprace, P. Harpe
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引用次数: 0

摘要

让$G$成为一个团体。一个子集$F \subset G$被称为不可约忠实的,如果存在一个不可约的酉表示$\pi$的$G$,使得$\pi(x) \ne \text{id}$对于所有$x \in F \smallsetminus \{e\}$。否则$F$被称为不可还原的不忠。给定一个正整数$n$,如果大小为$n$的每个子集都是不可约忠实的,我们说$G$具有属性$P(n)$。根据Gelfand和Raikov的经典结果,每个群体都有$P(1)$。Walter证明了每个组都有$P(2)$。很容易看出,有些组没有$P(3)$。给出了具有$P(n-1)$性质的(有限或无限)可数群$G$中大小为$n$的不可约不忠实子集的完整描述,证明了这样的子集包含在特定种类的有限初等阿贝尔正规子群$G$中。我们纯粹从群体结构的角度推导出属性$P(n)$的特征。由此可知,如果可数群$G$有$P(n-1)$而不有$P(n)$,则$n$是有限域上射影空间的基数。一个群$G$具有$Q(n)$的属性,如果对于每个不超过$n$大小的子集$F \subset G$,存在$G$的一个不可约的幺正表示$\pi$,使得$\pi(x) \ne \pi(y)$对于$F$中的任意一个不同的$x, y$。每个组都有$Q(2)$。对于可数群,可以看出Property $Q(3)$等价于$P(3)$, Property $Q(4)$等价于$P(6)$, Property $Q(5)$等价于$P(9)$。对于$m, n \ge 4$,属性$P(m)$和$Q(n)$之间的关系与加性组合学中一个记录良好的开放问题密切相关。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Groups with irreducibly unfaithful subsets for unitary representations
Let $G$ be a group. A subset $F \subset G$ is called irreducibly faithful if there exists an irreducible unitary representation $\pi$ of $G$ such that $\pi(x) \ne \text{id}$ for all $x \in F \smallsetminus \{e\}$. Otherwise $F$ is called irreducibly unfaithful. Given a positive integer $n$, we say that $G$ has Property $P(n)$ if every subset of size $n$ is irreducibly faithful. Every group has $P(1)$, by a classical result of Gelfand and Raikov. Walter proved that every group has $P(2)$. It is easy to see that some groups do not have $P(3)$. We provide a complete description of the irreducibly unfaithful subsets of size $n$ in a (finite or infinite) countable group $G$ with Property $P(n-1)$: it turns out that such a subset is contained in a finite elementary abelian normal subgroup of $G$ of a particular kind. We deduce a characterization of Property $P(n)$ purely in terms of the group structure. It follows that, if a countable group $G$ has $P(n-1)$ and does not have $P(n)$, then $n$ is the cardinality of a projective space over a finite field. A group $G$ has Property $Q(n)$ if, for every subset $F \subset G$ of size at most $n$, there exists an irreducible unitary representation $\pi$ of $G$ such that $\pi(x) \ne \pi(y)$ for any distinct $x, y$ in $F$. Every group has $Q(2)$. For countable groups, it is shown that Property $Q(3)$ is equivalent to $P(3)$, Property $Q(4)$ to $P(6)$, and Property $Q(5)$ to $P(9)$. For $m, n \ge 4$, the relation between Properties $P(m)$ and $Q(n)$ is closely related to a well-documented open problem in additive combinatorics.
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