Skip to main content
SpringerLink
Log in

Multiplicative Forms on Algebras and the Group Determinant

  • Chapter
  • First Online:
Group Matrices, Group Determinants and Representation Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2233))

  • 1167 Accesses

Abstract

The ideas in the initial papers by Frobenius on characters and group determinants are set out and put into context. It is indicated how the theory goes back to the search for “sums of squares identities”, the construction of “hypercomplex numbers” and the investigation of quadratic forms. The underlying objects, the group matrix, and its determinant, the group determinant, are introduced. It is shown that group matrices can be constructed as block circulants. The first construction by Frobenius of group characters for noncommutative groups is explained. This led to his construction of the irreducible factors of the group determinant. The k-characters, used to construct the irreducible factor corresponding to an irreducible character, are defined. Resulting developments are then discussed, with an indication of how the ideas of Frobenius were taken up by other mathematicians and how his approach and its continuation in the theory of norm forms on algebras have been useful. A summary of the various ways in which the work has impacted current areas is included.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J. Baez, The octonions. Bull. Am. Math Soc. 39, 145–205 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Bergmann, Formen auf Moduln über kommutativen Ringen beliebiger Charakteristik. J. Reine Angew. Math. 219, 113–156 (1965)

    MathSciNet  MATH  Google Scholar 

  3. A. Bergmann, Reduzierte Normen und Theorie von Algebren, in Algebra-Tagung Halle 1986, Tagungsband Wiss (Beiträge Halle (Saale) 1987/33 (M48) 1987), pp. 29–57

    Google Scholar 

  4. M. Bhargava, Higher composition laws I: a new view on Gauss composition, and quadratic generalizations. Ann. Math. 159, 217–250 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Bocher, Introduction to Higher Algebra (Dover, New York, 2004)

    Google Scholar 

  6. R. Bott, R.J. Milnor, On the parallelizability of the spheres. Bull. Am. Math. Soc. 64, 87–89 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  7. N. Bourbaki, Algebra I (Springer, Berlin, 1970)

    Google Scholar 

  8. H. Brandt, Der Kompositionsbegriff bei den quaterniären quadratischen Formen. Math. Ann 91, 300–315 (1924)

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Brandt, Über die Komponierbarheit der quaterniären quadratischen Formen. Math. Ann. 94, 179–197 (1925)

    Article  MathSciNet  Google Scholar 

  10. V.M. Buchstaber, E.G. Rees, The Gel’fand map and symmetric products. Sel. Math. (N.S.) 8, 523–535 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. V.M. Buchstaber, E.G. Rees, Rings of continuous functions, symmetric products and Frobenius Algebras. Russ. Math. Surv. 59, 125–144 (2004)

    Article  MathSciNet  Google Scholar 

  12. V.M. Buchstaber, E.G. Rees, Frobenius n-homomorphisms, transfers and branched coverings. Math. Proc. Camb. Philos. Soc. 144, 1–12 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. M.J. Collins, Modular analogues of Brauer’s characterisation of characters. J. Algebra 366, 35–41 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. J.H. Conway, D.A. Smith, On Quaternions and Octonians (A. K. Peters, Natick, 2003)

    Google Scholar 

  15. D. Cooper, G.S. Walsh, Three-manifolds, virtual homology, and group determinants. Geom. Topol. 10, 2247–2269 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. C.W. Curtis, Pioneers of Representation Theory (American Mathematical Society, Providence, 1999)

    MATH  Google Scholar 

  17. C.W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras. American Mathematical Society, 1962 (Chelsea Publishing, White River Junction, 2006)

    Google Scholar 

  18. P.J. Davis, Circulant Matrices (Chelsea, New York, 1994)

    MATH  Google Scholar 

  19. C. Deninger, On the analogue of the formula of Chowla and Selberg for real quadratic fields. J. Reine Angew. Math. 351, 171–191 (1984)

    MathSciNet  MATH  Google Scholar 

  20. P. Diaconis, Group Representations in Probability and Statistics (Institute of Mathematical Statistics, Hayward, 1988)

    Google Scholar 

  21. P. Diaconis, Patterned matrices. Matrix theory and applications (Phoenix, AZ, 1989), in Proceedings of Symposia in Applied Mathematics, vol. 40 (American Mathematical Society, Providence, RI, 1990), pp. 37–58

    Google Scholar 

  22. L.E. Dickson, On the group defined for any given field by the multiplication table of any given finite group. Trans. Am. Math. Soc. 3, 285–301 (1902)

    Article  MathSciNet  MATH  Google Scholar 

  23. L.E. Dickson, An elementary exposition of Frobenius’s theory of group-characters and group-determinants. Ann. Math. 4(2), 25–49 (1902)

    Article  MathSciNet  MATH  Google Scholar 

  24. L.E. Dickson, Modular theory of group matrices. Trans. Am. Math. Soc. 8, 389–398 (1907)

    Article  MathSciNet  MATH  Google Scholar 

  25. L.E. Dickson, Modular theory of group characters. Bull. Am. Math. Soc 13, 477–499 (1907)

    Article  MathSciNet  MATH  Google Scholar 

  26. L.E. Dickson, On quaternions and their generalization and the history of the eight square theorem. Ann. Math. 20(2) , 155–171 (1919)

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Dieudonné, Schur functions and group representations. Astérisque 87–88, 7–19 (1981)

    MathSciNet  MATH  Google Scholar 

  28. P.G.L. Dirichlet, Lectures on Number Theory, (Supplements by R. Dedekind). History of Mathematics, vol. 16 (American Mathematical Society, Providence, 1999)

    Google Scholar 

  29. E. Formanek, D. Sibley, The group determinant determines the group. Proc. Am. Math. Soc. 112, 649–656 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. G. Frobenius, Über Gruppencharaktere (Sitzungsber. Preuss. Akad. Wiss, Berlin, 1896), pp. 985–1021; Ges Abh. III, pp. 1–37

    Google Scholar 

  31. G. Frobenius, Über die Primfactoren der Gruppendeterminante (Sitzungsber Preuss. Akad. Wiss. Berlin, 1896), pp. 1343–1382; Ges Abh. III, pp. 38–77

    Google Scholar 

  32. W. Fulton, J. Harris, Representation Theory: A First Course. Graduate Texts in Mathematics, vol. 129 (Springer, New York, 1991) (Lecture 6)

    Google Scholar 

  33. P.X. Gallagher, Invariants for finite groups. Adv. Math. 34, 46–57 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  34. I.M. Gelfand, M. Kapranov, A. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston 1994)

    Book  MATH  Google Scholar 

  35. T. Hawkins, The origins of the theory of group characters. Arch. History Exact Sci. 7, 142–170 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  36. T. Hawkins, New light on Frobenius’ creation of the theory of group characters. Arch. History Exact Sci. 12, 17–243 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  37. T. Hawkins, Emergence of the theory of Lie groups, in An Essay in the History of Mathematics 1869–1926. Sources and Studies in the History of Mathematics and Physical Sciences (Springer, New York, 2000)

    Google Scholar 

  38. T. Hawkins, The Mathematics of Frobenius in Context (Springer, New York, 2013)

    Book  MATH  Google Scholar 

  39. H.-J. Hoehnke, K.W. Johnson, The 1-,2-, and 3-characters determine a group. Bull. Am. Math. Soc. 27, 243–245 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  40. H.-J. Hoehnke, K.W. Johnson, The 3-characters are sufficient for the group determinant, in Proceedings of the Second International Conference on Algebra. Contemporary Mathematics, vol. 184 (1995), pp. 193–206

    Google Scholar 

  41. H.-J. Hoehnke, K.W. Johnson, k-characters and group invariants. Commun. Algebra 26, 1–27 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  42. S.P. Humphries, Cogrowth of groups and the Dedekind-Frobenius group determinant. Math. Proc. Camb. Philos. Soc. 121, 193–217 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. S.P. Humphries, E.L. (Turner) Rode, Weak Cayley tables and generalized centralizer rings of finite groups. Math. Proc. Camb. Philos. Soc. 153, 281–318 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. T. Hurley, Group rings and rings of matrices. Int. J. Pure Appl. Math. 31, 319–335 (2006)

    MathSciNet  MATH  Google Scholar 

  45. T. Hurley, Convolutional codes from units in matrix and group rings. Int. J. Pure Appl. Math. 50, 431–463 (2009)

    MathSciNet  MATH  Google Scholar 

  46. T. Hurley, I. McLoughlin, A group ring construction of the extended binary Golay code. IEEE Trans. Inform. Theory 54, 4381–4383 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. I.M. Isaacs, Character Theory of Finite Groups (Academic, New York, 1976)

    MATH  Google Scholar 

  48. K.W. Johnson, Latin square determinants, in Algebraic, Extremal and Metric Combinatorics 1986. London Mathematical Society Lecture Notes Series, vol. 131 (1988), pp. 146–154

    Google Scholar 

  49. K.W. Johnson, On the group determinant. Math. Proc. Camb. Philos. Soc. 109, 299–311 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  50. K.W. Johnson, The Dedekind-Frobenius group determinant, new life in an old method, in Proceedings, Groups St Andrews 97 in Bath, II. London Mathematical Society Lecture Notes Series, vol. 261, (1999), pp. 417–428

    Google Scholar 

  51. K.W. Johnson, S.K. Sehgal, The 2-character table is not sufficient to determine a group. Proc. Am. Math. Soc. 119, 1021–1027 (1993)

    MATH  Google Scholar 

  52. K.W. Johnson, S.K. Sehgal, The 2-characters of a group and the group determinant. Eur. J. Comb. 16, 623–631 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  53. K.W. Johnson, J.D.H. Smith, Characters of finite quasigroups. Eur. J. Comb. 5, 43–50 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  54. K.W. Johnson, J.D.H. Smith, On the category of weak Cayley table morphisms between groups. Sel. Math. (N.S.) 13, 57–67 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  55. K.W. Johnson, P. Vojtěchovský, Right division in groups, Dedekind-Frobenius group matrices, and Ward quasigroups. Abh. Math. Semin. Univ. Hambg. 75, 121–136 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  56. K.W. Johnson, S. Mattarei, S.K. Sehgal, Weak Cayley tables. J. Lond. Math. Soc. 61, 395–411 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  57. I.L. Kantor, A.S. Sodolodnikov, Hypercomplex Numbers (Springer, Berlin, 1989)

    Book  Google Scholar 

  58. M. Kervaire, On the parallelizability of the spheres. Proc. Nat. Acad. Sci. U.S.A. 44, 280–283 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  59. H.M. Khudaverdian, T.T. Voronov, A short proof of the Buchstaber-Rees theorem. Philos. Trans. R. Soc. London, Ser. A 369, 1334–1345 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  60. W. Kimmerle, K.W. Roggenkamp, Non-isomorphic groups with isomorphic spectral tables and Burnside matrices. Chin. Ann. Math. Ser. B 15, 273–282 (1994)

    MathSciNet  MATH  Google Scholar 

  61. M.A. Knus, Quadratic forms, in Clifford Algebras and Spinors. Seminars in Mathematics, vol. 1 (Departamento de Matemática, University Campinas, Campinas, 1988)

    Google Scholar 

  62. S. Lang, in Cyclotomic Fields I and II. Combined, 2nd edn. Graduate Texts in Mathematics, vol. 121 (Springer, New York, 1990)

    Book  MATH  Google Scholar 

  63. P.G. Lejeune Dirichlet, Lectures on Number Theory (American Mathematical Society, Providence, 1999)

    Book  MATH  Google Scholar 

  64. G.W. Mackey, The Scope and History of Commutative and Noncommutative Harmonic Analysis. History of Mathematics, vol. 5 (American Mathematical Society, Providence; London Mathematical Society, London, 1992)

    Google Scholar 

  65. R. Mansfield, A group determinant determines its group. Proc Am. Math. Soc 116, 939–941 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  66. J. McKay, D. Sibley, Brauer pairs with the same 2-characters. Preprint

    Google Scholar 

  67. S. Okubo, Introduction to Octonian and Other Non-associative Algebras in Physics (Cambridge University Press, Cambridge, 1995)

    Book  MATH  Google Scholar 

  68. A. Pfister, Zur Darstellung von − 1 als Summe von Quadraten in einem Körper. J. Lond. Math. Soc. 40, 159–165 (1965)

    Article  MATH  Google Scholar 

  69. A. Pfister, Multiplikative quadratische Formen. Arch. Math. 16, 363–370 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  70. H. Poincaré, Sur les nombres complexes. Comptes Rendues Acad. Sci. Paris 99, 740–742 (1884), Oeuvres 5, 77–79

    Google Scholar 

  71. H. Poincaré, Sur l’intégration des équations linéaires et les périodes des intégrales abéliennes. J. des Math. Pures Appl. 9(5), 139–212 (1903), Oeuvres 3, 106–166

    Google Scholar 

  72. D.St.P. Richards, Algebraic methods toward higher order probability inequalities II. Ann. Probab. 32, 1509–1544 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  73. M. Roitman, A complete set of invariants for finite groups. Adv. Math. 41, 301–311 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  74. S. Sahi, Higher correlation inequalities. Combinatorica 28, 209–227 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  75. I. Schur, Neuer Begründung der Theorie der Gruppencharaktere (Sitzungsber. Akad. Wiss, Berlin, 1905), pp. 406–432; Ges. Abh I, 143–169

    Google Scholar 

  76. W.R. Scott, Half homomorphisms of groups. Proc. Am. Math. Soc. 8, 1141–1144 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  77. J.A. Sjogren, Connectivity and spectrum in a graph with a regular automorphism group of odd order. Internat. J. Algebra Comput. 4, 529–560 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  78. J.D.H. Smith, A left loop on the 15-sphere. J. Algebra 176, 128–138 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  79. J.D.H. Smith, An Introduction to Quasigroups and Their Representations (Chapman and Hall, London, 2006)

    Book  Google Scholar 

  80. A. Speiser, Gruppendeterminante und Körperdiskriminante. Math. Ann. 77, 546–562 (1916)

    Article  MathSciNet  Google Scholar 

  81. O. Taussky, Matrices of rational integers. Bull. Am. Math. Soc. 66, 327–345 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  82. O. Taussky, History of sums of squares in algebra. American mathematical heritage: algebra and applied mathematics (El Paso, Tex., 1975/Arlington, Tex., 1976), pp. 73–90, Math. Ser., 13, Texas Tech Univ., Lubbock, Tex., 1981

    Google Scholar 

  83. R.L. Taylor, Galois representations associated to Siegel modular forms of low weight. Duke Math. J. 63, 281–332 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  84. E. Trachtenberg, Singular value decomposition of Frobenius matrices for approximate and multi-objective signal processing tasks, in SVD and Signal Processing, ed. by E. Deprettere (1988), pp. 331–345

    Google Scholar 

  85. M.J. Vazirani, Extending Frobenius’ higher characters. Sci. Math. Jpn. 58, 169–182 (2003)

    MathSciNet  MATH  Google Scholar 

  86. M. Ward, Postulates for the inverse operations in a group. Trans. Am. Math. Soc. 32, 520–526 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  87. W.C. Waterhouse, Composition of norm-type forms. J. Reine Angew. Math. 353. 85–97 (1984)

    Google Scholar 

  88. H. Weber, Theorie der Abel’schen Zahlkörper I, section 3. Acta. Math. Bd 8, 193–263 (1886)

    Article  Google Scholar 

  89. H. Weber, Theorie der Abel’schen Zahlkörper IV, sections 2, 3. Acta. Math. Bd 9, 105–130 (1887)

    Article  MathSciNet  Google Scholar 

  90. H. Weber, Lehrbuch der Algebra, vol. III (Chelsea Publishing Company, White River Junction, 1961)

    Google Scholar 

  91. B. Weisfeiler, On Construction and Identification of Graphs. Springer Lecture Notes in Mathematics, vol. 558 (Springer, Berlin, 1976)

    Book  MATH  Google Scholar 

  92. A. Wiles, On ordinary λ-adic representations associated to modular forms. Invent. Math. 94, 529–573 (1988)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, Abington, PA, USA

    Kenneth W. Johnson

Authors
  1. Kenneth W. Johnson
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Johnson, K.W. (2019). Multiplicative Forms on Algebras and the Group Determinant. In: Group Matrices, Group Determinants and Representation Theory. Lecture Notes in Mathematics, vol 2233. Springer, Cham. https://doi.org/10.1007/978-3-030-28300-1_1

Download citation

Publish with us

Policies and ethics

Search

Navigation