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Further Group Matrices and Group Determinants

  • Kenneth W. Johnson
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2233)

Abstract

The construction and properties of group matrices are analyzed in more detail. The group matrix of a cyclic group with a certain ordering is a circulant matrix. The book by Davis (Circulant Matrices, Chelsea, New York, 1994) gives a comprehensive account of circulants and the chapter is designed to provide a far reaching extension and generalization of the results there. If an arbitrary subgroup H of a group G is taken, it is shown that with an appropriate ordering the group matrix XG is a block matrix in which each block is of the form \(X_{H}( \underline {u})\) where the elements in \( \underline {u}\) are all distinct and of the form \(x_{g_{i}}\). This leads to methods to partially diagonalize the group matrix which facilitate the calculation of the group determinant. Group matrix versions of the ring of representations and the Burnside ring of a group are described using supermatrices. A description of projective group matrices, corresponding to projective representations, is given. Work of L. E. Dickson on group matrices and group determinants over a finite field is also described.

References

  1. 4.
    S.A. Amitsur. Groups with representations of bounded degree. II. Ill. J. Math. 5, 198–205 (1961)MathSciNetCrossRefGoogle Scholar
  2. 8.
    J. Baez, The octonions. Bull. Am. Math Soc. 39, 145–205 (2002)MathSciNetCrossRefGoogle Scholar
  3. 14.
    F.A. Berezin, Introduction to Superanalysis (Reidel, Kufstein, 1987)CrossRefGoogle Scholar
  4. 33.
    R.A. Brualdi, Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir and Cayley. Linear Algebra Appl. 52/53, 769–791 (1983)MathSciNetCrossRefGoogle Scholar
  5. 49.
    P. Cartier, A course on determinants, in Conformal Invariance and String Theory, ed. by P. Dita, V. Georgescu (Academic, Boston, 1989), pp. 443–445CrossRefGoogle Scholar
  6. 51.
    P. Cartier, A primer on Hopf algebras. Preprint. IHES, IHES/M/06/40 (2006)Google Scholar
  7. 57.
    C. Cheng, A character theory for projective representations of finite groups. Linear Algebra Appl. 469, 30–242 (2015)MathSciNetCrossRefGoogle Scholar
  8. 58.
    C. Cheng, D. Han, On twisted group frames. Linear Algebra Appl. 569, 285–310 (2019)MathSciNetCrossRefGoogle Scholar
  9. 66.
    D. Cooper, G.S. Walsh, Three-manifolds, virtual homology, and group determinants. Geom. Topol. 10, 2247–2269 (2006)MathSciNetCrossRefGoogle Scholar
  10. 72.
    P.J. Davis, Circulant Matrices (Chelsea, New York, 1994)zbMATHGoogle Scholar
  11. 77.
    P. Deligne, J.W. Morgan, Notes on Supersymmetry (following Joseph Bernstein), in Quantum Fields and Strings: A Course for Mathematicians, vol. 1 (American Mathematical Society, Providence, 1999), pp. 41–97zbMATHGoogle Scholar
  12. 80.
    J. Deruyts, Essai d’une théorie générale des formes algébriques, Mém. Soc. R. Sci. Liège 17, 1–156 (1892)zbMATHGoogle Scholar
  13. 82.
    P. Diaconis, Group Representations in Probability and Statistics (Institute of Mathematical Statistics, Hayward, 1988)Google Scholar
  14. 89.
    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)MathSciNetCrossRefGoogle Scholar
  15. 91.
    L.E. Dickson, Modular theory of group matrices. Trans. Am. Math. Soc. 8, 389–398 (1907)MathSciNetCrossRefGoogle Scholar
  16. 92.
    L.E. Dickson, Modular theory of group characters. Bull. Am. Math. Soc 13, 477–499 (1907)MathSciNetCrossRefGoogle Scholar
  17. 99.
    W.B. Fite, Certain factors of the group determinant. Am. Math Mon. 13, 51–53 (1906)MathSciNetCrossRefGoogle Scholar
  18. 107.
    G. Frobenius, Über die Primfactoren der Gruppendeterminante (Sitzungsber Preuss. Akad. Wiss. Berlin, 1896), pp. 1343–1382; Ges Abh. III, pp. 38–77Google Scholar
  19. 109.
    G. Frobenius, Über Matrizen aus positiven Elementen (Sitzungsber Akad. Wiss. Berlin, 1908), pp. 471–476, Ges Abh. III, pp. 404–409Google Scholar
  20. 121.
    I.M. Gelfand, S. Gelfand, V. Retakh, R.L.Wilson, Quasideterminants. Adv. Math. 193, 56–141 (2005)CrossRefGoogle Scholar
  21. 122.
    M. Giuliani, K.W. Johnson, Right division in Moufang loops. Comment. Math. Univ. Carol. 51, 209–215 (2010)MathSciNetzbMATHGoogle Scholar
  22. 162.
    I.M. Isaacs, Character Theory of Finite Groups (Academic, New York, 1976)zbMATHGoogle Scholar
  23. 181.
    K.W. Johnson, J.D.H. Smith, Matched pairs, permutation representations, and the Bol property. Commun. Algebra 38, 2903–2914 (2010)MathSciNetCrossRefGoogle Scholar
  24. 188.
    G. Karpilovsky, Projective Representations of Finite Groups (Marcel Dekker, New York, Basel, 1985)zbMATHGoogle Scholar
  25. 200.
    S. Lang, in Cyclotomic Fields I and II. Combined, 2nd edn. Graduate Texts in Mathematics, vol. 121 (Springer, New York, 1990)CrossRefGoogle Scholar
  26. 229.
    D.V. Ouellette, Schur complements and statistics. Linear Algebra Appl. 36, 187–295 (1981)MathSciNetCrossRefGoogle Scholar
  27. 250.
    J.W. Sands, Base change for higher Stickelberger ideals. J. Number Theory 73, 518–526 (1998)MathSciNetCrossRefGoogle Scholar
  28. 251.
    I. Schur, Über die Darstellung der Endliche Gruppen durch gebrochenen lineare Substitutionen. J. reine angew. Math. 127, 20–50 (1904), Ges. Abh. I, 86–116Google Scholar
  29. 253.
    I. Schur, Untersuchungen über die Darstellung der Endliche Gruppen durch gebrochenen lineare Substitutionen. J. Reine Angew. Math. 132, 85–137 (1907); Ges. Abh. I, 198–250Google Scholar
  30. 258.
    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)MathSciNetCrossRefGoogle Scholar
  31. 269.
    R.P. Stanley, Invariants of finite groups and their relations to combinatorics. Bull. Am. Math. Soc. (New series) I, 475–511 (1979)Google Scholar
  32. 280.
    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. 331–345Google Scholar
  33. 283.
    V.S. Varadarajan, Supersymmetry for Mathematicians: An Introduction. Courant Lecture Notes in Mathematics, vol. 11 (American Mathematical Society, Providence, 2004)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Kenneth W. Johnson
    • 1
  1. 1.Department of MathematicsPennsylvania State UniversityAbingtonUSA

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