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.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Frame, J. S., "The Degrees of the Irreducible Representations of Simply Transitive Permutation Groups," Duke Math. Journal 3, (1937), 8–17.
Frame, J. S., "Group Decomposition by Double Coset Matrices," Bull. Amer. Math. Soc. 54, (1948) 740–755.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1978 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Frame, J.S. (1978). Orthogonal groups over GF(2) and related graphs. In: Alavi, Y., Lick, D.R. (eds) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol 642. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0070374
Download citation
DOI: https://doi.org/10.1007/BFb0070374
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-08666-6
Online ISBN: 978-3-540-35912-8
eBook Packages: Springer Book Archive
