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, Volume 12 (2020) no. 1, pp. 31-68.

Abstract

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.

Received: 2019-10-18
Accepted: 2020-01-04
Revised after acceptance: 2020-03-31
Published online: 2020-09-25

DOI: 10.5802/cml.61

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

Author's affiliations:

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

License:

CC-BY-NC-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Pierre-Emmanuel Caprace; Pierre de la Harpe. Groups with irreducibly unfaithful subsets for unitary representations. Confluentes Mathematici, Volume 12 (2020) no. 1, pp. 31-68. doi : 10.5802/cml.61. https://cml.centre-mersenne.org/articles/10.5802/cml.61/

References
Cited by

[1] K. Bauer, D. Sen, and P. Zvengrowski, A generalized Goursat lemma. Tatra Mt. Math. Publ. 64 (2015), 1–19. | DOI | MR | Zbl

[2] B. Bekka and P. de la Harpe, Irreducibly represented groups. Comment. Math. Helv. 83 (2008), 847–868. | DOI | MR | Zbl

[3] M. Bhargava, When is a group the union of proper normal subgroups? Amer. Math. Monthly 109 (2002), no. 5, 471–473. | DOI | MR | Zbl

[4] N. Bourbaki, Algèbre, Chapitres 1 à 3. Bourbaki, 1970.

[5] N. Bourbaki, Algèbre, Chapitre VIII, Modules et anneau semi-simples. Hermann, 1958. | DOI

[6] N. Bourbaki, Théories spectrales, Chapitres 1 et 2. Hermann, 1967. | DOI

[7] W. Burnside, The theory of groups of finite order, 2nd Edition. Cambridge Univ. Press, 1911. Reprint, Dover, 1955. | DOI | Zbl

[8] Y. Bugeaud and T.N. Shorey, On the Diophantine equation $\frac{x^{m} - 1}{x - 1} = \frac{y^{n} - 1}{y - 1}$ . Pacific J. Math. 207 (2002), no. 1, 61–75. | DOI | Zbl

[9] E.E. Croot, E. Samuel III, and V.F. Lev, Open problems in additive combinatorics. Amer. Math. Soc., Proc. Lecture Notes 43 (2007), 207–233. | DOI | Zbl

[10] J. Dixmier, Les C*-algèbres et leurs représentations, deuxième édition. Gauthier–Villars, 1969 [First Edition 1964, English translation North-Holland 1977].

[11] M. Fitch, and R. Jamison, Minimum sum covers of small cyclic groups. Proceedings of the Thirty-first Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 2000). Congr. Numer. 147 (2000), 65–81. | Zbl

[12] L. Fuchs, Infinite abelian groups. Vol. I. Academic Press, 1970. | Zbl

[13] W. Gaschütz, Endliche Gruppen mit treuen absolut–irreduziblen Darstellungen. Math. Nach. 12 (1954), 253–255. | DOI | MR | Zbl

[14] I. Gelfand and D. Raikov, Irreducible unitary representations of locally bicompact groups. Rec. Math. [Mat. Sbornik] N.S. 13(55) (1943), 301–316. I.M. Gelfand, Collected papers, Volume II, 3–17. | DOI | Zbl

[15] R. Goormaghtigh. L’intermédiaire des Mathématiciens 24 (1917), 88.

[16] F. Greenleaf and M. Moskowitz, Cyclic vectors for representations of locally compact groups. Math. Ann. 190 (1971), 265–288. | DOI | MR | Zbl

[17] P. de la Harpe, Topics in geometric group theory. The University of Chicago Press, 2000. | Zbl

[18] B. He, A remark on the Diophantine equation $\frac{x^{3} - 1}{x - 1} = \frac{y^{n} - 1}{y - 1}$ . Glas. Mat. Ser. III 44(64), no. 1 (2009), 1–6. | DOI | MR | Zbl

[19] B. Huppert, Character theory of finite groups. W. de Gruyter, 1998. | DOI | Zbl

[20] I.M. Isaacs, Character theory of finite groups. Academic Press, 1976. | Zbl

[21] E. Kowalski, An introduction to the representation theory of groups. Graduate Studies in Mathematics 155, Amer. Math. Soc., 2014. | DOI | Zbl

[22] J. Lambek, Lectures on rings and modules, Second Edition. Chelsea, 1976. [First Edition, Blaisdell, 1966.] | Zbl

[23] G.W. Mackey, The theory of unitary group representations. University of Chicago Press, 1976. | Zbl

[24] B.H. Neumann, Some remarks on infinite groups, J. London Math. Soc. 12 (1937), 120–127. | DOI | Zbl

[25] The On-Line Encyclopedia of Integer Sequences. On line: https://oeis.org | Zbl

[26] N. Radu, A classification theorem for boundary $2$ -transitive automorphisms of trees. Invent. Math. 209 (2017), no. 1, 1–60. | DOI | MR | Zbl

[27] R. Ratat. L’intermédiaire des Mathématiciens 23 (1916), 150.

[28] R. Remak, Über minimale invariante Untergruppen in der Theorie der endlichen Gruppen. J. reine angew. Math. 162 (1930), 1–16. | DOI | Zbl

[29] D.J.S. Robinson, A course in the theory of groups, Second Edition. Graduate Texts in Math. 80, Springer, 1996. | DOI

[30] Z. Sasvári, On a refinement of the Gel’fand–Raikov theorem. Math. Nachr. 150 (1991), 185–187. | DOI | MR | Zbl

[31] Z. Sasvári, Positive definite and definitizable functions. Academic Verlag, 1994. | Zbl

[32] D. Segal, Polycyclic groups. Cambridge Univ. Press, 1983. | DOI | Zbl

[33] K. Shoda, Bemerkungen über vollständig reduzible Gruppen. J. Fac. Sci., Univ. Tokyo, Sect. I 2 (1931), 203–209. | Zbl

[34] F. Szechtman, Groups having a faithful irreducible representation. J. Algebra 454 (2016), 292–307. | DOI | MR | Zbl

[35] M.E. Walter, A duality between locally compact groups and certain Banach algebras. J. Functional Analysis 17 (1974), 131–160. | DOI | MR | Zbl

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