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 FG is called irreducibly faithful if there exists an irreducible unitary representation π of G such that π(x)id for all xF{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 FG of size at most n, there exists an irreducible unitary representation π of G such that π(x)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,n4, 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 : 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
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     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/}
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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/item/CML_2020__12_1_31_0/

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