Skip to main content
SpringerLink
Log in

Realizations of abstract regular polytopes from a representation theoretic view

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

Peter McMullen has developed a theory of realizations of abstract regular polytopes, and has shown that the realizations up to congruence form a pointed convex cone which is the direct product of certain irreducible subcones. We show that each of these subcones is isomorphic to a set of positive semi-definite hermitian matrices of dimension m over either the real numbers, the complex numbers or the quaternions. In particular, we correct an erroneous computation of the dimension of these subcones by McMullen and Monson. We show that the automorphism group of an abstract regular polytope can have an irreducible character \({\chi}\) with \({\chi \neq \overline{\chi}}\) and with arbitrarily large essential Wythoff dimension. This gives counterexamples to a result of Herman and Monson, which was derived from the erroneous computation mentioned before. We also discuss a relation between cosine vectors of certain pure realizations and the spherical functions appearing in the theory of Gelfand pairs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Harmonic analysis on finite groups. In: Cambridge Studies in Advanced Mathematics, vol. 108. Cambridge University Press, Cambridge (2008). ISBN 978-0-521-88336-8. doi:10.1017/CBO9780511619823

  2. Cherkassoff, M., Sjerve, D.: On groups generated by three involutions, two of which commute. In: Mislin, G. (ed.) The Hilton Symposium 1993, vol. 6. CRM Proc. Lecture Notes, pp. 169–185. Amer. Math. Soc., USA (1994). http://www.math.ubc.ca/~sjer/3inv.pdf

  3. GAP.: GAP—groups, algorithms, and programming, version 4.7.6. The GAP Group (2014). http://www.gap-system.org

  4. Herman, A., Monson, B.: On the real Schur indices associated with infinite Coxeter groups. In: Yin Ho, C., Sin, P., Tiep, P.H., Turull, A. (ed.) Finite groups 2003, pp. 185–194. Walter de Gruyter, Berlin (2004)

  5. Howe, R.E.: On the character of Weil’s representation. Trans. Am. Math. Soc. 177:287–298 (1973). ISSN 0002-9947. http://www.jstor.org/stable/1996597

  6. Isaacs, I.M.: Characters of solvable and symplectic groups. Am. J. Math. 95(3), 594–635 (1973). doi:10.2307/2373731. http://www.jstor.org/stable/2373731

  7. Isaacs, I.M.: Character Theory of Finite Groups. Dover Publications, Inc., New York (1994). ISBN 0-486-68014-2 (corrected reprint)

  8. Macdonald, I.G.: Symmetric functions and Hall polynomials. In: Oxford Mathematical Monographs, 2 edn. Oxford University Press (Oxford Science Publications), Oxford, (1995). ISBN 0-19-853489-2 (With contributions by A. Zelevinsky)

  9. McMullen, P.: Realizations of regular polytopes. Aequationes Math. 37(1), 38–56 (1989). ISSN 0001-9054. doi:10.1007/BF01837943.

  10. McMullen, P.: Regular polyhedra related to projective linear groups. Discrete Math. 91(2), 161–170 (1991). ISSN 0012-365X. doi:10.1016/0012-365X(91)90107-D

  11. McMullen, P.: Realizations of regular polytopes, III. Aequationes Math. 82(1–2), 35–63 (2011). ISSN 0001-9054. doi:10.1007/s00010-010-0063-9

  12. McMullen, P.: Realizations of regular polytopes, IV. Aequat. Math. 87(1–2), 1–30 (2014). ISSN 0001-9054. doi:10.1007/s00010-013-0187-9

  13. McMullen, P., Monson, B.: Realizations of regular polytopes, II. Aequat. Math. 65(1–2), 102–112 (2003). ISSN 0001-9054. doi:10.1007/s000100300007

  14. McMullen, P., Schulte, E.: Abstract regular polytopes. In: Encyclopedia of Mathematics and its Applications, vol. 92. Cambridge University Press, Cambridge (2002). ISBN 0-521-81496-0. doi:10.1017/CBO9780511546686

  15. Prasad, A.: On character values and decomposition of the Weil representation associated to a finite abelian group. J. Anal. 17, 73–85 (2009). ISSN 0971-3611. http://arxiv.org/abs/0903.1486

  16. Simon, B.: Representations of finite and compact groups. In: Graduate Studies in Mathematics, vol. 10. American Mathematical Society, Providence (1996). ISBN 0-8218-0453-7

  17. Suzuki, M.: Group Theory I. In: Grundlehren der Mathematischen Wissenschaften, vol. 247. Springer, Berlin (1982). ISBN 3-540-10915-3 (Translated from the Japanese by the author)

  18. Thomas, T.: The character of the Weil representation. J. London Math. Soc. (2) 77(1), 221–239 (2008). ISSN 0024-6107. doi:10.1112/jlms/jdm098

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Rostock, Ulmenstr. 69, Haus 3, 18057, Rostock, Germany

    Frieder Ladisch

Authors
  1. Frieder Ladisch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frieder Ladisch.

Additional information

F. Ladisch was supported by the DFG (Project: SCHU 1503/6-1).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ladisch, F. Realizations of abstract regular polytopes from a representation theoretic view. Aequat. Math. 90, 1169–1193 (2016). https://doi.org/10.1007/s00010-016-0434-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-016-0434-y

Mathematics Subject Classification

Keywords

Search

Navigation