Groups with irreducibly unfaithful subsets for unitary representations

TL;DR

This paper characterizes irreducibly unfaithful subsets in groups with certain properties, linking group structure to the existence of specific irreducible unitary representations and exploring related properties in countable groups.

Contribution

It provides a complete description of unfaithful subsets in groups with Property P(n-1) and characterizes Property P(n) through group structure, connecting to finite projective spaces.

Findings

Unfaithful subsets are contained in specific finite elementary abelian subgroups.

Countable groups with P(n-1) but not P(n) relate to projective spaces over finite fields.

Relations between properties Q(n) and P(m) are linked to open problems in additive combinatorics.

Abstract

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 $\pi(x) \neq \mathrm{id}$ 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 $\pi$ 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.