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Generalized Iterated Wreath Products of Cyclic Groups and Rooted Trees Correspondence

  • Mee Seong Im
  • Angela Wu
Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 15)

Abstract

Consider the generalized iterated wreath product \(\mathbb {Z}_{r_1}\wr \mathbb {Z}_{r_2}\wr \ldots \wr \mathbb {Z}_{r_k}\) where \(r_i \in \mathbb {N}\). We prove that the irreducible representations for this class of groups are indexed by a certain type of rooted trees. This provides a Bratteli diagram for the generalized iterated wreath product, a simple recursion formula for the number of irreducible representations, and a strategy to calculate the dimension of each irreducible representation. We calculate explicitly fast Fourier transforms (FFT) for this class of groups, giving the literature’s fastest FFT upper bound estimate.

Keywords

Iterated wreath products Cyclic groups Rooted trees Irreducible representations Fast Fourier transform Bratteli diagrams 

AMS Subject Classification

Primary 20C99 20E08; Secondary 65T50 05E25 

Notes

Acknowledgments

The authors acknowledge Mathematics Research Communities for providing an exceptional working environment at Snowbird, Utah. They would like to thank Michael Orrison for helpful discussions, and the referees for extremely useful remarks on this manuscript. This manuscript was written during M.S.I.’s visit to the University of Chicago in 2014. She thanks their hospitality.

References

  1. 1.
    J.T. Astola, C. Moraga, R.S. Stanković, Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design (Wiley, Hoboken, 2005)Google Scholar
  2. 2.
    K. Balasubramanian, Enumeration of internal rotation reactions and their reaction graphs. Theor. Chim. Acta 53(2), 129–146 (1979)CrossRefGoogle Scholar
  3. 3.
    K. Balasubramanian, Graph theoretical characterization of NMR groups, nonrigid nuclear spin species and the construction of symmetry adapted NMR spin functions. J. Chem. Phys. 73(7), 3321–3337 (1980)MathSciNetCrossRefGoogle Scholar
  4. 4.
    D. Borsa, T. Graepel, A. Gordon, The wreath process: a totally generative model of geometric shape based on nested symmetries (2015). Preprint. arXiv:1506.03041Google Scholar
  5. 5.
    T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli, Clifford theory and applications. Functional analysis. J. Math. Sci. (N.Y.) 156(1), 29–43 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    W. Chang, Image processing with wreath product groups (2004), https://www.math.hmc.edu/seniorthesis/archives/2004/wchang/wchang-2004-thesis.pdf
  7. 7.
    A.J. Coleman, Induced Representations with Applications to S n and GL(n). Lecture notes prepared by C. J. Bradley. Queen’s Papers in Pure and Applied Mathematics, No. 4 (Queen’s University, Kingston, 1966)Google Scholar
  8. 8.
    J.W. Cooley, J.W. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)MathSciNetCrossRefGoogle Scholar
  9. 9.
    C.W. Curtis, I. Reiner, Methods of Representation Theory. Vol. I. Wiley Classics Library (Wiley, New York, 1990). With applications to finite groups and orders. Reprint of the 1981 original, A Wiley-Interscience PublicationGoogle Scholar
  10. 10.
    P. Diaconis, Average running time of the fast Fourier transform. J. Algorithms 1(2), 187–208 (1980)MathSciNetCrossRefGoogle Scholar
  11. 11.
    R. Foote, G. Mirchandani, D.N. Rockmore, D. Healy, T. Olson, A wreath product group approach to signal and image processing. I. Multiresolution analysis. IEEE Trans. Signal Process. 48(1), 102–132 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    R.B. Holmes, Mathematical foundations of signal processing II. the role of group theory. MIT Lincoln Laboratory, Lexington. Technical report 781 (1987), pp. 1–97Google Scholar
  13. 13.
    R.B. Holmes, Signal processing on finite groups. MIT Lincoln Laboratory, Lexington. Technical report 873 (1990), pp. 1–38Google Scholar
  14. 14.
    M.S. Im, A. Wu, Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence. Adv. Math. Sci. https://arxiv.org/abs/1409.0604 (to appear)
  15. 15.
    G. Karpilovsky, Clifford Theory for Group Representations. North-Holland Mathematics Studies, vol. 156 (North-Holland Publishing Co., Amsterdam, 1989) Notas de Matemática [Mathematical Notes], 125Google Scholar
  16. 16.
    M. Leyton, A Generative Theory of Shape, vol. 2145 (Springer, Berlin, 2003)zbMATHGoogle Scholar
  17. 17.
    D.K. Maslen, D.N. Rockmore, The Cooley-Tukey FFT and group theory. Not. AMS 48(10), 1151–1160 (2001)MathSciNetzbMATHGoogle Scholar
  18. 18.
    R. Milot, A.W. Kleyn, A.P.J. Jansen, Energy dissipation and scattering angle distribution analysis of the classical trajectory calculations of methane scattering from a Ni (111) surface. J. Chem. Phys. 115(8), 3888–3894 (2001)CrossRefGoogle Scholar
  19. 19.
    G. Mirchandani, R. Foote, D.N. Rockmore, D. Healy, T. Olson, A wreath product group approach to signal and image processing-part II: convolution, correlation, and applications. IEEE Trans. Signal Process. 48(3), 749–767 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    R.C. Orellana, M.E. Orrison, D.N. Rockmore, Rooted trees and iterated wreath products of cyclic groups. Adv. Appl. Math. 33(3), 531–547 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    L.R. Rabiner, R.W. Schafer, C.M. Rader, The chirp z-transform algorithm and its application. Bell Syst. Tech. J. 48, 1249–1292 (1969)MathSciNetCrossRefGoogle Scholar
  22. 22.
    C.M. Rader, Discrete Fourier transforms when the number of data samples is prime. Proc. IEEE 56(6), 1107–1108 (1968)CrossRefGoogle Scholar
  23. 23.
    D. Rockmore, Fast Fourier analysis for abelian group extensions. Adv. Appl. Math. 11(2), 164–204 (1990)MathSciNetCrossRefGoogle Scholar
  24. 24.
    M. Schnell, Understanding high-resolution spectra of nonrigid molecules using group theory. ChemPhysChem 11(4), 758–780 (2010)CrossRefGoogle Scholar

Copyright information

© The Author(s) and the Association for Women in Mathematics 2018

Authors and Affiliations

  • Mee Seong Im
    • 1
  • Angela Wu
    • 2
  1. 1.Department of Mathematical SciencesUnited States Military AcademyWest PointUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

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