Orthogonal groups over GF(2) and related graphs

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.