Subschemes of some association schemes. (English) Zbl 0762.20004

Bannai, Eiichi

J. Algebra 144, No. 1, 167-188 (1991).

Summary: By using the character tables of known commutative association schemes, we can, in many instances, construct some new commutative association schemes as their subschemes. In particular, we obtain many new non-group- case association schemes from the group association schemes of finite simple groups. Although examples worked out in this paper are limited to some special cases, it might be possible that this phenomenon occurs universally for all the group association schemes of finite Chevalley groups. The subschemes obtained in this way could be considered as an association scheme version of Weyl groups for Chevalley groups.

MSC:

20C15	Ordinary representations and characters
20C33	Representations of finite groups of Lie type
05E30	Association schemes, strongly regular graphs

Keywords:

character tables; commutative association schemes; finite simple groups; finite Chevalley groups; Weyl groups

References:

[1]	Bannai, E, Character tables of commutative association schemes, (), 105-128 · Zbl 0757.05100
[2]	Bannai, E; Ito, T, Algebraic combinatorics I, (1984), Benjamin/Cummings Menlo Park, CA · Zbl 0555.05019
[3]	Bannai, E; Hao, S; Song, S.Y, Character tables of the association schemes of finite orthogonal groups acting on the nonisotropic points, J. combin. theory ser. A, 54, 164-200, (1990) · Zbl 0762.20005
[4]	Bannai, E; Song, S.Y, The character tables of Paige’s simple Moufang loops and their relationship to the character tables of PSL(2, q), (), 209-236 · Zbl 0682.20050
[5]	Brouwer, A; Cohen, A; Neumaier, A, Distance-regular graphs, (1989), Springer-Verlag New York/Berlin · Zbl 0678.05038
[6]	Delsarte, P, An algebraic approach to the association schemes of coding theory, Philips res. rep. suppl., No. 10, (1973) · Zbl 1075.05606
[7]	Faradzev, I.A; Ivanov, A.A; Klin, M.H, Galois correspondence between permutation groups and cellular rings (association schemes), Graphs combin., 6, 303-332, (1990) · Zbl 0764.05099
[8]	Higman, D.G, Coherent configurations, Geom. dedicata, 4, 1-32, (1975) · Zbl 0333.05010
[9]	Ivanov, A.A; Muzichuk, M.E; Ustimenko, V.A, On a new family of (P and Q)-polynomial schemes, European J. combin., 10, 337-345, (1989) · Zbl 0709.05015
[10]	Liebeck, M.W; Praeger, C.E; Saxl, J, A classification of the maximal subgroups of the finite alternating and symmetric groups, J. algebra, 111, 365-383, (1987) · Zbl 0632.20011
[11]	Schur, I, Untersuchungen über die darstellungen der endlichen gruppen durch gebrochene lineare substitutionen, J. reine angew. math., 132, 85-137, (1907) · JFM 38.0174.02
[12]	Steinberg, R, The representations of GL(3, q), GL(4, q), PGL(3, q), and PGL(4. q), Canad. J. math., 3, 225-235, (1951) · Zbl 0042.25602
[13]	Suzuki, M, On a class of doubly transitive groups, Ann. of math., 75, 105-145, (1962) · Zbl 0106.24702