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Orthogonal groups over GF(2) and related graphs

  • J. Sutherland Frame
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 642)

Abstract

Regular graphs are considered, whose automorphism groups are permutation representations P of the orthogonal groups in various dimensions over GF(2). Vertices and adjacencies are defined by quadratic forms, and after graphical displays of the trivial isomorphisms between the symmetric groups S2, S3, S5, S6 and corresponding orthogonal groups, a 28-vertex graph is constructed that displays the isomorphism between S8 and o 6 + (2). Explored next are the eigenvalues and constituent idempotent matrices of the (−1,1)-adjacency matrix A of each of the orthogonal graphs, and the commuting ring R of the rank three permutation representation P of its automorphism group. Formulas are obtained for splitting into its irreducible characters χ(i) the permutation character χ of P, by expressing the class sums Bλ of P in terms of the identity matrix and the (0,1)-matrices H and K obtained from the adjacency matrix A=H − K.

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References

  1. 1.
    Frame, J. S., "The Degrees of the Irreducible Representations of Simply Transitive Permutation Groups," Duke Math. Journal 3, (1937), 8–17.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Frame, J. S., "Group Decomposition by Double Coset Matrices," Bull. Amer. Math. Soc. 54, (1948) 740–755.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Seidel, J. S., "On Two-Graphs and Shult's Characterization of Symplectic and orthogonal Geometries Over GF(2)," T.H. — Report 73-WSK-02, Technological University, Eindhoven, The Netherlands.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1978

Authors and Affiliations

  • J. Sutherland Frame
    • 1
  1. 1.Michigan State UniversityEast LansingUSA

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