具有酉表示不可约不忠实子集的群
Groups with irreducibly unfaithful subsets for unitary representations
摘要：
让$G$成为一个组。一个子集$F \subset G$称为不可约忠实的，如果存在$G$的不可约酉表示$π$，使得对所有$x \in F \smallsetminus \{e\}$的$π(x) \neq \mathrm{id}$.否则，$F$被称为不可还原的不忠。给定一个正整数$n$，我们说$G$具有性质$P(n)$，如果每个大小为$n$的子集都是不可约忠实的。根据Gelfand和Raikov的一个经典结果，每个群都有$P(1)$. Walter证明了每个群都有$P(2)$。很容易看出有些组没有$P(3)$。我们提供了一个完整的描述不可约的不忠实子集的大小$n$在可数群$G$（有限或无限）的性质$P(n-1)$：它原来这样的子集包含在一个有限的初等阿贝尔正规子群的$G$的一个特定的种类。我们推导出性质$P(n)$的一个表征纯粹的组结构。因此，如果一个可数群$G$有$P(n-1)$且没有$P(n)$，则$n$是有限域上的射影空间的基数。一个群$G$具有性质$Q(n)$，如果对于每个大小至多为$n$的子集$F \subset G$，存在$G$的不可约酉表示$π$，使得对于$F$中的任何不同的$x, y$，$π(x) \ne π(y)$。每个组都有$Q(2)$。对于可数群，证明了性质$Q(3)$等价于$P(3)$，性质$Q(4)$等价于$P(6)$，性质$Q(5)$等价于$P(9)$。对于$m, n \ge 4$，属性$P(m)$和$Q(n)$之间的关系与加法组合学中一个有据可查的开放问题密切相关。
英文摘要：
Let $G$ be a group. A subset $F \subset G$ is called irreducibly faithful if there exists an irreducible unitary representation $π$ of $G$ such that $π(x) \neq \mathrm{id}$ 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 countable group $G$ (finite or infinite) 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 $π$ of $G$ such that $π(x) \ne π(y)$ 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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