## Characters of finite quasigroups. VI: Critical examples and doubletons.(English)Zbl 0704.20056

[Part V see ibid. 10, No.5, 449-456 (1989; Zbl 0679.20059).]
The aim of this paper is to present critical examples of scheme character tables which are demonstrably not quasigroup character tables. Immediate sporadic examples are offered by the character tables of the schemes given by strongly regular graphs with 26 vertices. The main task of the note is to present members of an infinite family of examples, such as $\left[\begin{matrix} 1 & 1 & 1 \\ \sqrt{7} & 1/3\sqrt{7} & -1/3\sqrt{7} \\ 2\sqrt{5} & -1/3\sqrt{5} & 2/15\sqrt{5}\end{matrix}\right]$ of scheme character tables which are demonstrably not quasigroup character tables. This is done in Theorem: Let r be an even integer greater than 4. Then the character table of the Johnson scheme $$J(r,2)$$ is not a quasigroup character table.
Reviewer: C.Pereira da Silva

### MSC:

 20N05 Loops, quasigroups 05E30 Association schemes, strongly regular graphs 20C99 Representation theory of groups

Zbl 0679.20059
Full Text:

### References:

 [1] Johnson, K.W.; Smith, J.D.H., Characters of finite quasigroups, Europ. J. combin., 5, 43-50, (1984) · Zbl 0537.20042 [2] Johnson, K.W.; Smith, J.D.H., Characters of finite quasigroups II: induced characters, Europ. J. combin., 7, 131-137, (1986) · Zbl 0599.20110 [3] Johnson, K.W.; Smith, J.D.H., Characters of finite quasigroups III: quotients and fusion, Europ. J. combin., 10, 47-56, (1989) · Zbl 0667.20053 [4] Johnson, K.W.; Smith, J.D.H., Characters of finite quasigroups IV: products and superschemes, Europ. J. combin., 10, 257-263, (1989) · Zbl 0669.20053 [5] Johnson, K.W.; Smith, J.D.H., Characters of finite quasigroups V: linear characters, Europ. J. combin., 10, 449-456, (1989) · Zbl 0679.20059 [6] Albert, A.A., Quasigroups II, Trans. am. math. soc., 55, 401-409, (1944) · Zbl 0063.00042 [7] Weisfeiler, B., On construction and identification of graphs, Springer lecture notes in mathematics no. 558, (1976), Springer-Verlag Berlin [8] Bannai, E.; Ito, T., Current research on algebraic combinatorics, Graphs combin., 2, 287-308, (1968) · Zbl 0685.05030 [9] Bannai, E.; Ito, T., Algebraic combinatorics (in Russian), (1987), Mir Moscow [10] E. Bannai, S. Hao and S. Y. Song, Character tables of the association schemes of finite orthogonal groups acting on the nonisotropic points, J. Combin. Th. (A), to appear. · Zbl 0762.20005 [11] Song, S.Y., The character tables of certain association schemes, ph.D. dissertation, (1987), Ohio State University, Columbus Ohio [12] Paige, L.J., A class of simple Moufang loops, Proc. am. math. soc., 7, 471-482, (1956) · Zbl 0070.25302 [13] Baer, R., Nets and groups, Trans. am. math. soc., 46, 110-141, (1939) · JFM 65.0819.02 [14] Johnson, K.W., S-rings over loops, right mapping groups and transversals in permutation groups, Math. proc. camb. phil. soc., 89, 433-443, (1981) · Zbl 0462.20058 [15] Smith, J.D.H., Representation theory of infinite groups and finite quasigroups, Séminare de mathématiques supérieures, (1986), Université de Montréal Montréal [16] O’Nan, M.E., Sharply 2-transitive sets of permutations, () · Zbl 0639.20002 [17] Johnson, K.W., Loop transversals and the centralizer ring of a permutation group, Math. proc. camb. phil. soc., 94, 411-416, (1983) · Zbl 0597.20002 [18] Nomura, K., On t-homogeneous permutation sets, Archiv math., 44, 485-487, (1985) · Zbl 0547.20003 [19] Bannai, E.; Ito, T., Algebraic combinatorics I: association schemes, mathematics lecture notes no. 58, (1984), Benjamin-Cummings Menlo Park, California [20] Delsarte, P., An algebraic approach to the association schemes of coding theory, Philips res. repts., 10, Suppl., (1973) · Zbl 1075.05606 [21] Bose, R.C.; Shimamoto, T., Classification and analysis of partially balanced designs with two associate classes, J. amer. statist. assoc., 47, 151-190, (1952) · Zbl 0048.11603 [22] Shrikhande, S.S., On a characterization of the triangular association scheme, Ann. math. statist., 30, 39-47, (1959) · Zbl 0089.15101 [23] Connor, W.S., The uniqueness of the triangular association scheme, Ann. math. statist., 29, 262-266, (1958) · Zbl 0085.35601 [24] Terwilliger, P., The Johnson graph J(d, r) is unique if (d, r) ≠ (2, 8), Discr. math., 58, 175-189, (1986) · Zbl 0587.05038 [25] Klin, M.Ch.; Pöschel, R.; Rosenbaum, K., Angewandte algebra, (1988), VEB Deutscher Verlag der Wissenschaften Berlin · Zbl 0648.20001 [26] Diaconis, P., Group representation in probability and statistics, IMS lecture notes-monograph series no. 11, (1988), Institute of Mathematical Statistics Hayward, California
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.