On representation zeta functions of groups and a conjecture of Larsen–Lubotzky

Avni, Nir; Klopsch, Benjamin; Onn, Uri; Voll, Christopher

doi:10.1016/j.crma.2010.02.019

Cited by 16 publications

(47 citation statements)

References 10 publications

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“…The expression for the representation zeta polynomial of GL 4 (O 1 ) is rather long, so we omit the details here (see Steinberg [28]). Among the other centralizers appearing in this table, only the results regarding the representations of group G (2,1,1) are not clear from our discussion so far. We follow a method of Uri Onn to discuss the representations of these groups.…”

Section: Representations Of Gl 3 (O 2 )mentioning

confidence: 83%

“…In particular, we discuss the relation between the representation zeta polynomial of GL n (O 2 ) and those of centralizers in GL n (O 1 ). We also construct all the irreducible representations of groups GL 2 …”

Section: Applicationsmentioning

confidence: 99%

“…ρ χ ρ χ i , and as dim ρ χ = dim ρ χ i = q 2 , we conclude that ρ χ i is an extension of ρ χ to G (2,1,1) . By the Clifford theory, it follows that all the representations of G (2,1,1) that lie above ρ χ are of the…”

mentioning

confidence: 91%

“…We claim that the representation ρ χ can be extended to G (2,1,1) . Let H i = Z (H) × O 1 × O 1 be the pre-image in H of the maximal isotropic subgroup O 1 × O 1 for the above bilinear form.…”

mentioning

confidence: 98%

“…Hence the contribution to the representation zeta polynomial of G (2,1,1) from these representations is…”

mentioning

confidence: 98%

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On representations of general linear groups over principal ideal local rings of length two

Singla

2010

Journal of Algebra

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We study the irreducible complex representations of general linear groups over principal ideal local rings of length two with a fixed finite residue field. We construct a canonical correspondence between the irreducible representations of all such groups that preserves dimensions. For general linear groups of order three and four over these rings, we construct all the irreducible representations. We show that the problem of constructing all the irreducible representations of all general linear groups over these rings is not easier than the problem of constructing all the irreducible representations of all general linear groups over principal ideal local rings of arbitrary length in the function field case.

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Section: Representations Of Gl 3 (O 2 )mentioning

confidence: 83%

Section: Applicationsmentioning

confidence: 99%

mentioning

confidence: 91%

mentioning

confidence: 98%

“…Hence the contribution to the representation zeta polynomial of G (2,1,1) from these representations is…”

mentioning

confidence: 98%

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On representations of general linear groups over principal ideal local rings of length two

Singla

2010

Journal of Algebra

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Counting irreducible representations of the Heisenberg group over the integers of a quadratic number field

Ezzat

2014

Journal of Algebra

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We calculate the representation zeta function of the Heisenberg group over the integers of a quadratic number field. In general, the representation zeta function of a finitely generated torsion-free nilpotent group enumerates equivalence classes of representations, called twist-isoclasses. This calculation is based on an explicit description of a representative from each twist-isoclass. Our method of construction involves studying the eigenspace structure of the elements of the image of the representation and then picking a suitable basis for the underlying vector space.

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Representation zeta functions of some compact 𝑝-adic analytic groups

Avni¹,

Klopsch²,

Onn³

et al. 2012

Zeta Functions in Algebra and Geometry

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Using the Kirillov orbit method, novel methods from p-adic integration and Clifford theory, we study representation zeta functions associated to compact p-adic analytic groups. In particular, we give general estimates for the abscissae of convergence of such zeta functions.We compute explicit formulae for the representation zeta functions of some compact p-adic analytic groups, defined over a compact discrete valuation ring o of characteristic 0. These include principal congruence subgroups of SL2(o), without any restrictions on the residue field characteristic of o, as well as the norm one group SL1(D) of a nonsplit quaternion algebra D over the field of fractions of o and its principal congruence subgroups. We also determine the representation zeta functions of principal congruence subgroups of SL3(o) in the case that o has residue field characteristic 3 and is unramified over Z3.Theorem 1.4. Let o be a compact discrete valuation ring of characteristic 0, with residue field of cardinality q and characteristic 3. Suppose that o is unramified over Z 3 . Then, for all m ∈ N, one hasThe four theorems are proved and discussed in Sections 2.3, 3.1, 4.2 and 5 respectively. Whilst all of the results in the current paper are of a local nature, they might be best appreciated in the 'global' context of representation zeta functions of arithmetic groups, as explained to some extent below and exposed, in much greater detail, in [2,3,4].

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On representation zeta functions of groups and a conjecture of Larsen–Lubotzky

Cited by 16 publications

References 10 publications

On representations of general linear groups over principal ideal local rings of length two

On representations of general linear groups over principal ideal local rings of length two

Counting irreducible representations of the Heisenberg group over the integers of a quadratic number field

Representation zeta functions of some compact 𝑝-adic analytic groups

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