Advertisement

Generalized Iterated Wreath Products of Symmetric Groups and Generalized 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 \(S_{r_1}\wr \ldots \wr S_{r_k}\) of symmetric groups. We give a complete description of the traversal for the generalized iterated wreath product. We also prove an existence of a bijection between the equivalence classes of ordinary irreducible representations of the generalized iterated wreath product and orbits of labels on certain rooted trees. We find a recursion for the number of these labels and the degrees of irreducible representations of the generalized iterated wreath product. Finally, we give rough upper bound estimates for fast Fourier transforms.

Keywords

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

AMS Subject Classification

Primary 20C30 20E08; Secondary 65T50 05E18 05E10 

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 immensely valuable comments. This paper 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.
    T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli, Representation Theory of the Symmetric Groups. Cambridge Studies in Advanced Mathematics, vol. 121 (Cambridge University Press, Cambridge, 2010). The Okounkov-Vershik approach, character formulas, and partition algebrasGoogle Scholar
  7. 7.
    T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli, Representation Theory and Harmonic Analysis of Wreath Products of Finite Groups. London Mathematical Society Lecture Note Series, vol. 410 (Cambridge University Press, Cambridge, 2014)Google Scholar
  8. 8.
    W. Chang, Image processing with wreath product groups (2004), https://www.math.hmc.edu/seniorthesis/archives/2004/wchang/wchang-2004-thesis.pdf
  9. 9.
    M. Clausen, U. Baum, Fast Fourier transforms for symmetric groups: theory and implementation. Math. Comput. 61(204), 833–847 (1993)MathSciNetCrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    J.W. Cooley, J.W. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)MathSciNetCrossRefGoogle Scholar
  12. 12.
    K.-D. Crisman, M.E. Orrison, Representation theory of the symmetric group in voting theory and game theory, in Algebraic and Geometric Methods in Discrete Mathematics. Contemporary Mathematics, vol. 685 (American Mathematical Society, Providence, 2017), pp. 97–115Google Scholar
  13. 13.
    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
  14. 14.
    Z. Daugherty, A.K. Eustis, G. Minton, M.E. Orrison, Voting, the symmetric group, and representation theory. Am. Math. Mon. 116(8), 667–687 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    W. Fulton, J. Harris, Representation Theory. Graduate Texts in Mathematics, vol. 129 (Springer, New York, 1991). A first course, Readings in MathematicsGoogle Scholar
  17. 17.
    T. Geetha, A. Prasad, Comparison of Gelfand-Tsetlin bases for alternating and symmetric groups (2017). Preprint. arXiv:1606.04424Google Scholar
  18. 18.
    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
  19. 19.
    R.B. Holmes, Signal processing on finite groups. MIT Lincoln Laboratory, Lexington. Technical report 873 (1990), pp. 1–38Google Scholar
  20. 20.
    M.S. Im, A. Wu, Generalized iterated wreath products of cyclic groups and rooted trees correspondence. Adv. Math. Sci., https://arxiv.org/abs/1409.0603 (to appear)
  21. 21.
    G. James, A. Kerber, The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and Its Applications, vol. 16 (Addison-Wesley Publishing Co., Reading, 1981). With a foreword by P.M. Cohn, With an introduction by Gilbert de B. RobinsonGoogle Scholar
  22. 22.
    G. Karpilovsky, Clifford Theory for Group Representations. North-Holland Mathematics Studies, vol. 156 (North-Holland Publishing Co., Amsterdam, 1989). Notas de Matemática [Mathematical Notes], 125CrossRefGoogle Scholar
  23. 23.
    M.G. Karpovsky, E.A. Trachtenberg, Fourier transform over finite groups for error detection and error correction in computation channels. Inf. Control 40(3), 335–358 (1979)MathSciNetCrossRefGoogle Scholar
  24. 24.
    A. Kerber, Representations of Permutation Groups. I. Lecture Notes in Mathematics, vol. 240 (Springer, Berlin, 1971)CrossRefGoogle Scholar
  25. 25.
    A. Kleshchev, Representation theory of symmetric groups and related Hecke algebras. Bull. Am. Math. Soc. 47(3), 419–481 (2010)MathSciNetCrossRefGoogle Scholar
  26. 26.
    S. Lee, Understanding voting for committees using wreath products (2010), https://www.math.hmc.edu/seniorthesis/archives/2010/slee/slee-2010-thesis.pdf
  27. 27.
    M. Leyton, A Generative Theory of Shape, vol. 2145 (Springer, Berlin, 2003)zbMATHGoogle Scholar
  28. 28.
    D.K. Maslen, The efficient computation of Fourier transforms on the symmetric group. Math. Comput. 67(223), 1121–1147 (1998)MathSciNetCrossRefGoogle Scholar
  29. 29.
    D.K. Maslen, D.N. Rockmore, The Cooley-Tukey FFT and group theory. Not. AMS 48(10), 1151–1160 (2001)MathSciNetzbMATHGoogle Scholar
  30. 30.
    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
  31. 31.
    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
  32. 32.
    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
  33. 33.
    D.N. Rockmore, Fast Fourier transforms for wreath products. Appl. Comput. Harmon. Anal. 2(3), 279–292 (1995)MathSciNetCrossRefGoogle Scholar
  34. 34.
    M. Schnell, Understanding high-resolution spectra of nonrigid molecules using group theory. ChemPhysChem 11(4), 758–780 (2010)CrossRefGoogle Scholar
  35. 35.
    B. Simon, Representations of Finite and Compact Groups. Graduate Studies in Mathematics, vol. 10 (American Mathematical Society, Providence, 1996)Google Scholar
  36. 36.
    A.M. Vershik, A.Y. Okounkov, A new approach to the representation theory of the symmetric groups. II. Zapiski Nauchnykh Seminarov POMI 307, 57–98 (2004)Google 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

Personalised recommendations