Groups with irreducibly unfaithful subsets for unitary representations

Pierre-Emmanuel Caprace; Pierre de la Harpe

doi:10.5802/cml.61

Pierre-Emmanuel Caprace ¹ ; Pierre de la Harpe ²

¹ UCLouvain – IRMP, Chemin du Cyclotron 2, box L7.01.02, B-1348 Louvain-la-Neuve
² Section de mathématiques, Université de Genève, C.P. 64, CH-1211 Genève 4

Confluentes Mathematici, Tome 12 (2020) no. 1, pp. 31-68.

Résumé

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 $π (x) \neq id$ for all $x \in F ∖ {e}$ . 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 $π (x) \neq π (y)$ 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 $m, n \geq 4$ , the relation between Properties $𝒫 (m)$ and $𝒬 (n)$ 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 : 10.5802/cml.61

Classification : 43A65, 22D10
Mots clés : Countable group, unitary representation, irreducible representation, faithful representation, factor representation, finite elementary abelian normal subgroup

Affiliations des auteurs :

Pierre-Emmanuel Caprace ¹ ; Pierre de la Harpe ²

¹ UCLouvain – IRMP, Chemin du Cyclotron 2, box L7.01.02, B-1348 Louvain-la-Neuve
² Section de mathématiques, Université de Genève, C.P. 64, CH-1211 Genève 4

Licence :

CC-BY-NC-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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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/articles/10.5802/cml.61/

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