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Examples of simply reducible groups. (English) Zbl 1477.20027

A simply reducible group \(G\) is a group having the following properties. 1. Every element of \(G\) is conjugate to its inverse, 2. Tensor product of any two irreducible representations of \(G\) decomposes into a direct sum of irreducible representations of \(G\) with multiplicities 0 or 1. These groups have important applications in physic and chemistry. The author of the paper under review presents by study and analysis of the structure and representations of such groups some new examples of simply reducible groups such as dihedral groups, some point groups, some dicyclic groups, generalized quaternion groups, Heisenberg groups over prime field of characteristic 2, some Cliford groups, and some Coxeter groups. The author also gives the precise decompositions of products of irreducible characters of dihedral groups, Heisenberg groups, and some Coxeter groups, giving the Clebsch-Gordan coefficients for these groups. The paper is very informative with a long list of references.

MSC:

20C35 Applications of group representations to physics and other areas of science
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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[1] ahulpke,Use GAP to define finite Clifford group, Mathematics Stack Exchange, URL: https://math.stackexchange.com/q/3213387(version: 2019-05-04), ahulpke’s homepage in Mathematics Stack Exchange:https://math.stackexchange.com/users/159739/ ahulpke
[2] G. E. Andrews,The Theory of Partitions, reprint of the 1976 original, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998.
[3] I. Armeanu,About the Schur index for ambivalent groups, Comm. Algebra42(2014), no. 2, 540-544.https://doi.org/10.1080/00927872.2012.717436 · Zbl 1290.20005
[4] M. Artin,Algebra, second edition, Pearson Education, Inc., 2011, English reprint edition published by PEARSON EDUCATION ASIA LTD. and CHINA MACHINE PRESS.
[5] M. Aschbacher, R. Lyons, S. D. Smith, and R. Solomon,The classification of finite simple groups, Mathematical Surveys and Monographs,172, American Mathematical Society, Providence, RI, 2011.https://doi.org/10.1090/surv/172
[6] R. Ayoub,An Introduction to the Analytic Theory of Numbers, Mathematical Surveys, No. 10, American Mathematical Society, Providence, RI, 1963. · Zbl 0128.04303
[7] M. Baake,Structure and representations of the hyperoctahedral group, J. Math. Phys. 25(1984), no. 11, 3171-3182.https://doi.org/10.1063/1.526087 · Zbl 0551.20008
[8] J. L. Berggren,Finite groups in which every element is conjugate to its inverse, Pacific J. Math.28(1969), 289-293.http://projecteuclid.org/euclid.pjm/1102983447 · Zbl 0172.03101
[9] Y. G. Berkovich, L. S. Kazarin, and E. M. Zhmud’,Characters of finite groups. Vol. 2, second edition, De Gruyter Expositions in Mathematics, vol. 64, De Gruyter, Berlin, 2019. · Zbl 1426.20003
[10] H. Bottomley,The number of partitions ofn,https://oeis.org/A000041, 2001, The On-Line Encyclopedia of Integer Sequences (OEIS), A000041 , [Online; accessed 16February-2020].
[11] N. Bourbaki,Lie groups and Lie algebras. Chapters 4-6, translated from the 1968 French original by Andrew Pressley, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. · Zbl 0983.17001
[12] R. W. Carter,Conjugacy classes in the Weyl group, Compositio Math.25(1972), 1-59. · Zbl 0254.17005
[13] ,Finite groups of Lie type, John Wiley & Sons, Ltd., 1985.
[14] T. Ceccherini-Silberstein, F. Scarabotti, and F. Tolli,Harmonic analysis on finite groups, Cambridge Studies in Advanced Mathematics,108, Cambridge University Press, Cambridge, 2008.https://doi.org/10.1017/CBO9780511619823 · Zbl 1149.43001
[15] ,Mackey’s theory ofτ-conjugate representations for finite groups, Jpn. J. Math. 10(2015), no. 1, 43-96.https://doi.org/10.1007/s11537-014-1390-8 · Zbl 1346.20006
[16] J.-Q. Chen, J. Ping, and F. Wang,Group representation theory for physicists, second edition, World Scientific Publishing Co., Inc., River Edge, NJ, 2002.https://doi.org/ 10.1142/5019 · Zbl 1015.20012
[17] L. Christine Kinsey and T. E. Moore,Symmetry, shape, and space. An introduction to mathematics through geometry, Key College Publishing, 2002. · Zbl 0974.00002
[18] H. S. M. Coxeter and W. O. J. Moser,Generators and relations for discrete groups, third edition, Springer-Verlag, New York, 1972. · Zbl 0239.20040
[19] ,Generators and relations for discrete groups, fourth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas],14, Springer-Verlag, Berlin-New York, 1980.
[20] L. Dornhoff,Group representation theory. Part A:Ordinary representation theory, Marcel Dekker, Inc., New York, 1971. · Zbl 0227.20002
[21] M. S. Dresselhaus, G. Dresselhaus, and A. Jorio,Group theory, Springer-Verlag, Berlin, 2008. · Zbl 1175.20001
[22] D. Faddeev and I. Sominski,Problems in higher algebra, Mir Publishers, Moscow, 1972, Translated from the Russian by George Yankovsky, Revised from the 1968 Russian edition.
[23] H. G. Feichtinger, W. Kozek, and F. Luef,Gabor analysis over finite abelian groups, Appl. Comput. Harmon. Anal.26(2009), no. 2, 230-248.https://doi.org/10.1016/ j.acha.2008.04.006 · Zbl 1162.43002
[24] J. S. Frame,The classes and representations of the groups of27lines and28bitangents, Ann. Mat. Pura Appl. (4)32(1951), 83-119.https://doi.org/10.1007/BF02417955
[25] ,The characters of the Weyl groupE8, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), pp. 111-130, Pergamon, Oxford, 1970.
[26] GAP Centres,GAP - Groups, Algorithms, Programming,https://www.gap-system. org, Version GAP 4.10.1 released on 23 February 2019.
[27] M. Geck and G. Pfeiffer,Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series,21, The Clarendon Press, Oxford University Press, New York, 2000. · Zbl 0996.20004
[28] A. Gelessus,Character tables for chemically important point groups,http://symmetry. jacobs-university.de/, 2019, [Online; accessed 16-August-2019].
[29] A. Gelessus, W. Thiel, and W. Weber,Multipoles and symmetry, J. Chemical Education 72(1995), no. 6, 505-508.
[30] J. B. Geloun and S. Ramgoolam,Tensor models, Kronecker coefficients and permutation centralizer algebras, J. High Energ. Phys.92(2017).https://doi.org/10.1007/ jhep11(2017)092 · Zbl 1383.81213
[31] S. Givant and P. Halmos,Introduction to Boolean algebras, Undergraduate Texts in Mathematics, Springer, New York, 2009.https://doi.org/10.1007/978-0-387-684369 · Zbl 1168.06001
[32] J. Goss,Molecular examples for point groups,https://www.staff.ncl.ac.uk/j.p. goss/symmetry/Molecules_pov.html, 2019, [Online; accessed 16-August-2019].
[33] D. Gross,Finite phase space methods in quantum information,https://www.qc.unifreiburg.de/files/diplom.pdf, 2005, Diploma thesis, Universit¨at Potsdam, [Online; accessed 25-August-2019].
[34] L. C. Grove,The characters of the hecatonicosahedroidal group, J. Reine Angew. Math. 265(1974), 160-169.https://doi.org/10.1515/crll.1974.265.160 · Zbl 0275.20015
[35] B. C. Hall,Quantum theory for mathematicians, Graduate Texts in Mathematics,267, Springer, New York, 2013.https://doi.org/10.1007/978-1-4614-7116-5 · Zbl 1273.81001
[36] M. Hamermesh,Group theory and its application to physical problems, Addison-Wesley Series in Physics, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1962. · Zbl 0100.36704
[37] S. D. Howard, A. R. Calderbank, and W. Moran,The finite Heisenberg-Weyl groups in radar and communications, EURASIP J. Appl. Signal Process.2006(2006), Art. ID 85685, 12 pp.https://doi.org/10.1155/asp/2006/85685 · Zbl 1122.94015
[38] J.-S. Huang,Lectures on representation theory, World Scientific Publishing Co., Inc., River Edge, NJ, 1999.https://doi.org/10.1142/9789812815743 · Zbl 0947.22001
[39] J. E. Humphreys,Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics,29, Cambridge University Press, Cambridge, 1990.https://doi.org/10. 1017/CBO9780511623646 · Zbl 0725.20028
[40] T. Inui, Y. Tanabe, and Y. Onodera,Group theory and its applications in physics, Springer Series in Solid-State Sciences,78, Springer-Verlag, Berlin, 1990, Translated and revised from the 1980 Japanese edition by the authors. Softcover reprint of the hardcover 1st edition 1990. · Zbl 0711.20009
[41] I. M. Isaacs,Algebra, Brooks/Cole Publishing Co., Pacific Grove, CA, 1994.
[42] G. James and A. Kerber,The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1985. First published by Addison-Wesley in 1981; digitally printed version 2009. · Zbl 0491.20010
[43] G. James and M. Liebeck,Representations and characters of groups, second edition, Cambridge University Press, New York, 2001.https://doi.org/10.1017/ CBO9780511814532 · Zbl 0981.20004
[44] D. L. Johnson,Presentations of groups, second edition, London Mathematical Society Student Texts,15, Cambridge University Press, Cambridge, 1997.https://doi.org/ 10.1017/CBO9781139168410 · Zbl 0906.20019
[45] D. L. Johnson and E. F. Robertson,Finite groups of deficiency zero, in Homological group theory (Proc. Sympos., Durham, 1977), 275-289, London Math. Soc. Lecture Note Ser.,36, Cambridge University Press, Cambridge, 1979.
[46] R. Kane,Reflection groups and invariant theory, CMS Books in Mathematics/Ouvrages de Math´ematiques de la SMC,5, Springer-Verlag, New York, 2001.https://doi.org/ 10.1007/978-1-4757-3542-0
[47] L. S. Kazarin and E. I. Chankov,Finite simply reducible groups are solvable, Sb. Math. 201(2010), no. 5-6, 655-668; translated from Mat. Sb.201(2010), no. 5, 27-40.https: //doi.org/10.1070/SM2010v201n05ABEH004087 · Zbl 1205.20008
[48] L. S. Kazarin and V. V. Yanishevski˘ı,On finite simply reducible groups, St. Petersburg Math. J.19(2008), no. 6, 931-951; translated from Algebra i Analiz19(2007), no. 6, 86-116.https://doi.org/10.1090/S1061-0022-08-01028-5
[49] E. I. Khukhro and V. D. Mazurov,Unsolved problems in group theory, The Kourovka Notebook, arXiv e-prints (2014), arXiv:1401.0300, Submitted on 1 Jan 2014 (v1), last revised 26 Mar 2019 (v16). · Zbl 1372.20001
[50] M. Koca, R. Ko¸c, M. Al-Barwani, and S. Al-Farsi,Maximal subgroups of the Coxeter groupW(H4)and quaternions, Linear Algebra Appl.412(2006), no. 2-3, 441-452. https://doi.org/10.1016/j.laa.2005.07.018 · Zbl 1087.20036
[51] M. Ladd,Symmetry of crystals and molecules, Oxford University Press, Oxford, 2014. https://doi.org/10.1093/acprof:oso/9780199670888.001.0001 · Zbl 1343.00024
[52] H. B. Lawson, Jr., and M.-L. Michelsohn,Spin geometry, Princeton Mathematical Series, 38, Princeton University Press, Princeton, NJ, 1989.
[53] G. W. Mackey,Multiplicity free representations of finite groups, Pacific J. Math.8 (1958), 503-510.http://projecteuclid.org/euclid.pjm/1103039895 · Zbl 0161.02301
[54] G. E. Martin,Transformation geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1982.
[55] G. A. Miller,Note on the groups generated by operators transforming each other into their inverses, Quart. J.40(1909), 366-367, contained in The collected works of George Abram Miller, Volume III,https://hdl.handle.net/2027/mdp.39015040409990. · JFM 40.0189.01
[56] ,Second note on the groups generated by operators transforming each other into their inverses [A correction], Quart. J.44(1913), 142-146, contained in The collected works of George Abram Miller, Volume III,https://hdl.handle.net/2027/mdp. 39015040409990. · JFM 44.0166.01
[57] M. Misaghian,The representations of the Heisenberg group over a finite field, Armen. J. Math.3(2010), no. 4, 162-173. · Zbl 1281.20052
[58] U. M¨uller,Symmetry relationships between crystal structures:applications of crystallographic group theory in crystal chemistry, International Union of Crystallography Texts on Crystallography,18, Oxford University Press, Oxford, 2013.
[59] D. Mumford,Tata lectures on theta. III, reprint of the 1991 original, Modern Birkh¨auser Classics, Birkh¨auser Boston, Inc., Boston, MA, 2007.https://doi.org/10.1007/9780-8176-4578-6
[60] I. Pak, G. Panova, and D. Yeliussizov,Bounds on the largest Kronecker and induced multiplicities of finite groups, Comm. Algebra47(2019), no. 8, 3264-3279.https: //doi.org/10.1080/00927872.2018.1555837 · Zbl 1477.20016
[61] J. Patera and R. Twarock,Affine extension of noncrystallographic Coxeter groups and quasicrystals, J. Phys. A35(2002), no. 7, 1551-1574.https://doi.org/10.1088/03054470/35/7/306 · Zbl 1012.20045
[62] E. W. Read,Projective characters of the Weyl group of typeF4, J. London Math. Soc. (2)8(1974), 83-93.https://doi.org/10.1112/jlms/s2-8.1.83 · Zbl 0279.20010
[63] M. S. Richman, T. W. Parks, and R. G. Shenoy,Discrete-time, discrete-frequency, timefrequency analysis, IEEE Transactions on Signal Processing46(1998), no. 6, 1517-1527. · Zbl 1010.94526
[64] H. Robbins,A remark on Stirling’s formula, Amer. Math. Monthly62(1955), 26-29. https://doi.org/10.2307/2308012 · Zbl 0068.05404
[65] S. Roman,Fundamentals of group theory, Birkh¨auser/Springer, New York, 2012.https: //doi.org/10.1007/978-0-8176-8301-6 · Zbl 1244.20001
[66] H. E. Rose,A course on finite groups, Universitext, Springer-Verlag London, Ltd., London, 2009.https://doi.org/10.1007/978-1-84882-889-6 · Zbl 1200.20001
[67] J. J. Rotman,An introduction to the theory of groups, fourth edition, Graduate Texts in Mathematics,148, Springer-Verlag, New York, 1995.https://doi.org/10.1007/9781-4612-4176-8 · Zbl 0810.20001
[68] B. E. Sagan,The symmetric group, second edition, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001.https://doi.org/10.1007/978-1-4757-6804-6 · Zbl 0964.05070
[69] SageMath licensed under the GPL,SageMath,http://www.sagemath.org/index.html, Version SageMath 8.7 released on 23 March 2019.
[70] D. E. Sands,Introduction to crystallography, Dover Publications, Inc., New York, 1993, this Dover edition, first published in 1993, is an unabridged republication of the work first published by W. A. Benjamin, Inc. (in the “Physical Chemistry Monograph Series”of the “Advanced Book Program”), 1969 (corrected printing 1975).
[71] F. Sasaki, M. Sekiya, T. Noro, K. Ohtsuki, and Y. Osanai,Non-relativistic configuration interaction calculations for many-electron atoms:Atomci, Modern Techniques in Computational Chemistry: MOTECC-91 (E. Clementi, ed.), Springer Netherlands, 1991.
[72] E. M. Schmidt,Number of conjugacy classes in Weyl group of typeDn,https://oeis. org/A234254, 2013, The On-Line Encyclopedia of Integer Sequences (OEIS), A234254, [Online; accessed 16-February-2020].
[73] J.-P. Serre,Finite groups:an introduction, Surveys of Modern Mathematics, vol. 10, International Press, Somerville, MA; Higher Education Press, Beijing, 2016, With assistance in translation provided by Garving K. Luli and Pin Yu.
[74] B. Simon,Representations of finite and compact groups, Graduate Studies in Mathematics,10, American Mathematical Society, Providence, RI, 1996. · Zbl 0840.22001
[75] R. Tao,Group theory in physics, Chinese ed., Higher Education Press, 2011.
[76] T. Tokuyama,On the decomposition rules of tensor products of the representations of the classical Weyl groups, J. Algebra88(1984), no. 2, 380-394.https://doi.org/10. 1016/0021-8693(84)90072-3
[77] J. Tolar,A classification of finite quantum kinematics, J. Physics: Conference Series 538(2014), 012020.
[78] A. J. van Zanten and E. de Vries,Criteria for groups with representations of the second kind and for simple phase groups, Canadian J. Math.27(1975), no. 3, 528-544.https: //doi.org/10.4153/CJM-1975-064-4 · Zbl 0338.20010
[79] E. P. Wigner,On representations of certain finite groups, Amer. J. Math.63(1941), 57-63.https://doi.org/10.2307/2371276 · JFM 67.0073.03
[80] Wikipedia contributors,Stone-von Neumann theorem — Wikipedia, the free encyclopedia,https://en.wikipedia.org/w/index.php?title=Stone-von_Neumann_theorem& oldid=932501700, 2019, [Online; accessed 17-January-2020].
[81] ,Molecular symmetry — Wikipedia,the free encyclopedia,https://en. wikipedia.org/w/index.php?title=Molecular_symmetry&oldid=962044761, 2020, [Online; accessed 13-June-2020].
[82] H. S. Wilf,Lectures on integer partitions, 2000, PIMS Distinguished Chair Lecture Notes, available inhttp://www.mathtube.org/lecture/notes/lectures-integerpartitions.
[83] P. Woit,Quantum mechanics for mathematicians:The Heisenberg group and the Schr¨odinger representation,http://citeseerx.ist.psu.edu/viewdoc/download?doi= 10.1.1.419.4136&rep=rep1&type=pdf, 2012, [Online; accessed 25-August-2019].
[84] Wolfram Research,Wolfram Mathematica, License purchased by the university, Version 11.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.