# CONFLUENTES MATHEMATICI

Groups with irreducibly unfaithful subsets for unitary representations
Confluentes Mathematici, Tome 12 (2020) no. 1, pp. 31-68.

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 $\pi \left(x\right)\ne \mathrm{id}$ for all $x\in F\setminus \left\{e\right\}$. Otherwise $F$ is called irreducibly unfaithful. Given a positive integer $n$, we say that $G$ has Property $𝒫\left(n\right)$ if every subset of size $n$ is irreducibly faithful. Every group has $𝒫\left(1\right)$, by a classical result of Gelfand and Raikov. Walter proved that every group has $𝒫\left(2\right)$. It is easy to see that some groups do not have $𝒫\left(3\right)$.

We provide a complete description of the irreducibly unfaithful subsets of size $n$ in a countable group $G$ (finite or infinite) with Property $𝒫\left(n-1\right)$: 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 $𝒫\left(n\right)$ purely in terms of the group structure. It follows that, if a countable group $G$ has $𝒫\left(n-1\right)$ and does not have $𝒫\left(n\right)$, then $n$ is the cardinality of a projective space over a finite field.

A group $G$ has Property $𝒬\left(n\right)$ if, for every subset $F\subset G$ of size at most $n$, there exists an irreducible unitary representation $\pi$ of $G$ such that $\pi \left(x\right)\ne \pi \left(y\right)$ for any distinct $x,y$ in $F$. Every group has $𝒬\left(2\right)$. For countable groups, it is shown that Property $𝒬\left(3\right)$ is equivalent to $𝒫\left(3\right)$, Property $𝒬\left(4\right)$ to $𝒫\left(6\right)$, and Property $𝒬\left(5\right)$ to $𝒫\left(9\right)$. For $m,n\ge 4$, the relation between Properties $𝒫\left(m\right)$ and $𝒬\left(n\right)$ is closely related to a well-documented open problem in additive combinatorics.

Reçu le : 2019-10-18
Accepté le : 2020-01-04
Accepté après révision le : 2020-03-31
Publié le : 2020-09-25
DOI : https://doi.org/10.5802/cml.61
Classification : 43A65,  22D10
Mots clés: Countable group, unitary representation, irreducible representation, faithful representation, factor representation, finite elementary abelian normal subgroup
@article{CML_2020__12_1_31_0,
author = {Pierre-Emmanuel Caprace and Pierre de la Harpe},
title = {Groups with irreducibly unfaithful subsets for unitary representations},
journal = {Confluentes Mathematici},
pages = {31--68},
publisher = {Institut Camille Jordan},
volume = {12},
number = {1},
year = {2020},
doi = {10.5802/cml.61},
language = {en},
url = {cml.centre-mersenne.org/item/CML_2020__12_1_31_0/}
}
Pierre-Emmanuel Caprace; Pierre de la Harpe. Groups with irreducibly unfaithful subsets for unitary representations. Confluentes Mathematici, Tome 12 (2020) no. 1, pp. 31-68. doi : 10.5802/cml.61. https://cml.centre-mersenne.org/item/CML_2020__12_1_31_0/

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