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Representation zeta function of a family of maximal class groups: Non-exceptional primes

  • Shannon Ezzat ORCID logo EMAIL logo
Published/Copyright: March 8, 2024

Abstract

We use a constructive method to obtain all but finitely many 𝑝-local representation zeta functions of a family of finitely generated nilpotent groups M n with maximal nilpotency class. For representation dimensions coprime to all primes p < n , we construct all irreducible representations of M n by defining a standard form for the matrices of these representations.

Acknowledgements

The author would like to thank Ben Martin and Christopher Voll for helpful and illuminating discussion, and the reviewers for their thoughtful feedback and suggestions.

  1. Communicated by: Benjamin Klopsch

References

[1] N. Avni, Arithmetic groups have rational representation growth, Ann. of Math. (2) 174 (2011), no. 2, 1009–1056. 10.4007/annals.2011.174.2.6Search in Google Scholar

[2] N. Avni, B. Klopsch, U. Onn and C. Voll, On representation zeta functions of groups and a conjecture of Larsen–Lubotzky, C. R. Math. Acad. Sci. Paris 348 (2010), no. 7–8, 363–367. 10.1016/j.crma.2010.02.019Search in Google Scholar

[3] N. Avni, B. Klopsch, U. Onn and C. Voll, Arithmetic groups, base change, and representation growth, Geom. Funct. Anal. 26 (2016), no. 1, 67–135. 10.1007/s00039-016-0359-6Search in Google Scholar

[4] L. Bartholdi and P. de la Harpe, Representation zeta functions of wreath products with finite groups, Groups Geom. Dyn. 4 (2010), no. 2, 209–249. 10.4171/ggd/81Search in Google Scholar

[5] M. N. Berman, I. Glazer and M. M. Schein, Pro-isomorphic zeta functions of nilpotent groups and Lie rings under base extension, Trans. Amer. Math. Soc. 375 (2022), no. 2, 1051–1100. 10.1090/tran/8506Search in Google Scholar

[6] D. A. Craven, Lower bounds for representation growth, J. Group Theory 13 (2010), no. 6, 873–890. 10.1515/jgt.2010.029Search in Google Scholar

[7] M. du Sautoy and L. Woodward, Zeta Functions of Groups and Rings, Lecture Notes in Math. 1925, Springer, Berlin, 2008. 10.1007/978-3-540-74776-5Search in Google Scholar

[8] S. Ezzat, Representation growth of finitely generated torsion-free nilpotent groups: Methods and examples, Ph.D. Thesis, University of Canterbury, 2012. Search in Google Scholar

[9] S. Ezzat, Counting irreducible representations of the Heisenberg group over the integers of a quadratic number field, J. Algebra 397 (2014), 609–624. 10.1016/j.jalgebra.2013.08.028Search in Google Scholar

[10] S. Ezzat, Representation growth of maximal class groups: Various exceptional cases, preprint (2014), https://arxiv.org/abs/1410.4992. Search in Google Scholar

[11] E. Hrushovski, B. Martin and S. Rideau, Definable equivalence relations and zeta functions of groups, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 10, 2467–2537. 10.4171/jems/817Search in Google Scholar

[12] A. Jaikin-Zapirain, Zeta function of representations of compact 𝑝-adic analytic groups, J. Amer. Math. Soc. 19 (2006), no. 1, 91–118. 10.1090/S0894-0347-05-00501-1Search in Google Scholar

[13] B. Klopsch, Representation growth and representation zeta functions of groups, Note Mat. 33 (2013), no. 1, 107–120. Search in Google Scholar

[14] A. Lubotzky and A. R. Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 336 (1985), no. 58, 1–117. 10.1090/memo/0336Search in Google Scholar

[15] A. Lubotzky and B. Martin, Polynomial representation growth and the congruence subgroup problem, Israel J. Math. 144 (2004), 293–316. 10.1007/BF02916715Search in Google Scholar

[16] C. Nunley and A. Magid, Simple representations of the integral Heisenberg group, Classical Groups and Related Topics (Beijing 1987), Contemp. Math. 82, American Mathematical Society, Providence (1989), 89–96. 10.1090/conm/082/982280Search in Google Scholar

[17] T. Rossmann, Computing topological zeta functions of groups, algebras, and modules, I, Proc. Lond. Math. Soc. (3) 110 (2015), no. 5, 1099–1134. 10.1112/plms/pdv012Search in Google Scholar

[18] T. Rossmann, Computing local zeta functions of groups, algebras, and modules, Trans. Amer. Math. Soc. 370 (2018), no. 7, 4841–4879. 10.1090/tran/7361Search in Google Scholar

[19] R. Snocken, Zeta functions of groups and rings, Ph.D. Thesis, University of Southampton, 2014. Search in Google Scholar

[20] A. Stasinski and C. Voll, Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type 𝐡, Amer. J. Math. 136 (2014), no. 2, 501–550. 10.1353/ajm.2014.0010Search in Google Scholar

[21] C. Voll, Functional equations for zeta functions of groups and rings, Ann. of Math. (2) 172 (2010), no. 2, 1181–1218. 10.4007/annals.2010.172.1181Search in Google Scholar

[22] E. Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), no. 1, 153–209. 10.1007/BF02100009Search in Google Scholar

[23] M. Zordan, Univariate and bivariate zeta functions of unipotent group schemes of type 𝐺, Internat. J. Algebra Comput. 32 (2022), no. 4, 653–682. 10.1142/S0218196722500291Search in Google Scholar

Received: 2022-12-22
Revised: 2024-02-09
Published Online: 2024-03-08
Published in Print: 2024-09-01

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