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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J.T. Astola, C. Moraga, R.S. Stanković, Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design (Wiley, Hoboken, 2005)
K. Balasubramanian, Enumeration of internal rotation reactions and their reaction graphs. Theor. Chim. Acta 53(2), 129–146 (1979)
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)
D. Borsa, T. Graepel, A. Gordon, The wreath process: a totally generative model of geometric shape based on nested symmetries (2015). Preprint. arXiv:1506.03041
T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli, Clifford theory and applications. Functional analysis. J. Math. Sci. (N.Y.) 156(1), 29–43 (2009)
W. Chang, Image processing with wreath product groups (2004), https://www.math.hmc.edu/seniorthesis/archives/2004/wchang/wchang-2004-thesis.pdf
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)
J.W. Cooley, J.W. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)
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 Publication
P. Diaconis, Average running time of the fast Fourier transform. J. Algorithms 1(2), 187–208 (1980)
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)
R.B. Holmes, Mathematical foundations of signal processing II. the role of group theory. MIT Lincoln Laboratory, Lexington. Technical report 781 (1987), pp. 1–97
R.B. Holmes, Signal processing on finite groups. MIT Lincoln Laboratory, Lexington. Technical report 873 (1990), pp. 1–38
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)
G. Karpilovsky, Clifford Theory for Group Representations. North-Holland Mathematics Studies, vol. 156 (North-Holland Publishing Co., Amsterdam, 1989) Notas de MatemĂ¡tica [Mathematical Notes], 125
M. Leyton, A Generative Theory of Shape, vol. 2145 (Springer, Berlin, 2003)
D.K. Maslen, D.N. Rockmore, The Cooley-Tukey FFT and group theory. Not. AMS 48(10), 1151–1160 (2001)
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)
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)
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)
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)
C.M. Rader, Discrete Fourier transforms when the number of data samples is prime. Proc. IEEE 56(6), 1107–1108 (1968)
D. Rockmore, Fast Fourier analysis for abelian group extensions. Adv. Appl. Math. 11(2), 164–204 (1990)
M. Schnell, Understanding high-resolution spectra of nonrigid molecules using group theory. ChemPhysChem 11(4), 758–780 (2010)
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 The Author(s) and the Association for Women in Mathematics
About this paper
Cite this paper
Im, M.S., Wu, A. (2018). Generalized Iterated Wreath Products of Cyclic Groups and Rooted Trees Correspondence. In: Deines, A., Ferrero, D., Graham, E., Im, M., Manore, C., Price, C. (eds) Advances in the Mathematical Sciences. AWMRS 2017. Association for Women in Mathematics Series, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-98684-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-98684-5_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-98683-8
Online ISBN: 978-3-319-98684-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)