Abstract
For a few decades the smallest known non-Schurian coherent configuration was the association scheme on 15 points, coming from a doubly regular tournament. Last year the second author, using a computer, enumerated all coherent configurations of order up to 15. A consequence of the enumeration is that all coherent configurations up to 13 points are Schurian and a unique non-Schurian rank 11 coherent configuration of order 14 exists. This coherent configuration has two fibers of sizes 6 and 8, and an automorphism group of order 24 isomorphic to SL(2, 3). We provide a computer free interpretation of this new object, relying on some simple interplay between group theoretical and combinatorial arguments.
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References
Bannai E., Ito T.: Algebraic Combinatorics. I: Association schemes. The Benjamin/Cummings Publishing, Menlo Park (1984).
Biggs N.: Algebraic Graph Theory, 2nd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1993).
Bose R.C., Shimamoto T.: Classification and analysis of partially balanced incomplete block designs with two associate classes. J. Am. Stat. Assoc. 47, 151–184 (1952).
Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 18. Springer-Verlag, Berlin (1989).
Cameron P.J.: Combinatorics: Topics, Techniques, Algorithms. Cambridge University Press, Cambridge (1994).
Coxeter H.S.M.: Regular Complex Polytopes. Cambridge University Press, London (1974).
Coxeter H.S.M., Moser W.O.J.: Generators and Relations for Discrete Groups, 3rd edn. Springer-Verlag, New York (1972). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14.
Dixon J.D., Mortimer B.: Permutation Groups. Graduate Texts in Mathematics, vol. 163. Springer-Verlag, New York (1996).
Evdokimov S., Kovács I., Ponomarenko I.: Characterization of cyclic Schur groups. Algebra i Analiz 25(5), 61–85 (2013).
Faradžev I.A., Klin M.H.: Computer package for computations with coherent configurations. In: Proceedings of the ISSAC, pp. 219–223. ACM Press, Bonn (1991).
Faradžev I.A., Ivanov A.A., Klin M.H.: Galois correspondence between permutation groups and cellular rings (association schemes). Graphs Comb. 6(4), 303–332 (1990).
Faradžev I.A., Klin M.H., Muzichuk M.E.: Cellular rings and groups of automorphisms of graphs. In: Investigations in Algebraic Theory of Combinatorial Objects. Mathematics and Its Application (Soviet Seties), vol. 84, pp. 1–152. Kluwer Academic Publishers, Dordrecht (1994).
The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4.12 (2008).
Hanaki A., Miyamoto I.: Classification of association schemes with 16 and 17 vertices. Kyushu J. Math. 52(2), 383–395 (1998).
Harary F.: Graph Theory. Addison-Wesley, Reading (1969).
Hardy G.H., Wright E.M.: An Introduction to the Theory of Numbers, 4th edn. Clarendon Press, Oxford (1960).
Higman D.G.: Coherent configurations. I. Rend. Sem. Mat. Univ. Padova 44, 1–25 (1970).
Higman D.G.: Strongly regular designs and coherent configurations of type \(\left[\begin{array}{ll} 3 \quad 2 \\ \quad \; 3 \\ \end{array}\right]\). Eur. J. Comb. 9(4), 411–422 (1988).
Hulpke A.: Constructing transitive permutation groups. J. Symb. Comput. 39(1), 1–30 (2005).
Kalužnin L.A., Beleckij P.M., Fejnberg V.Z.: Kranzprodukte. Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 101. BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1987) (with English, French and Russian summaries).
Klin M., Muzychuk M., Pech C., Woldar A., Zieschang P.-H.: Association schemes on 28 points as mergings of a half-homogeneous coherent configuration. Eur. J. Comb. 28(7), 1994–2025 (2007).
Klin M.Ch., Pöschel R., Rosenbaum K.: Angewandte Algebra für Mathematiker und Informatiker. Friedr. Vieweg & Sohn, Braunschweig (1988). Einführung in gruppentheoretisch-kombinatorische Methoden [Introduction to group-theoretical combinatorial methods].
Klin M., Pech C., Reichard S., Woldar A., Ziv-Av M.: Examples of computer experimentation in algebraic combinatorics. Ars Math. Contemp. 3(2), 237–258 (2010).
Klin M.H., Reichard S.: Construction of small strongly regular designs. Tr. Inst. Mat. Mekh. 19(3), 164–178 (2013).
McKay B.D.: Nauty User’s Guide (version 1.5) (1990).
Miyamoto I., Hanaki A.: Classification of association schemes with small vertices. http://math.shinshu-u.ac.jp/~hanaki/as/ (2014).
Nagatomo, A., Shigezumi, J.: Coherent configurations with at most 13 vertices. http://researchmap.jp/mu45vd02h-1782674 (2009).
Pasechnik D.V.: Skew-symmetric association schemes with two classes and strongly regular graphs of type \(L_{2n-1}(4n-1)\). Acta Appl. Math. 29(1–2), 129–138 (1992). Interactions between algebra and combinatorics.
Pöschel R.: Untersuchungen von \(S\)-Ringen, insbesondere im Gruppenring von \(p\)-Gruppen. Math. Nachr. 60, 1–27 (1974).
Schur I.: Zur theorie der einfach transitiven permutationsgruppen. Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., pp. 598–623 (1933).
See K., Song S.Y.: Association schemes of small order. J. Stat. Plan. Inference 73(1–2), 225–271 (1998). R. C. Bose Memorial Conference (Fort Collins, CO, 1995).
Shrikhande S.S.: The uniqueness of the \(L_{2}\) association scheme. Ann. Math. Stat. 30, 781–798 (1959).
Soicher L.H.: GRAPE: a system for computing with graphs and groups. In: Groups and Computation (New Brunswick, NJ, 1991). DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, vol. 11, pp. 287–291. American Mathematical Society, Providence (1993).
Weisfeiler B.: On Construction and Identification of Graphs. Lecture Notes in Mathematics, vol. 558. Springer, Berlin (1976).
Wielandt H.: Finite Permutation Groups. Translated from the German by R. Bercov. Academic Press, New York (1964).
Ziv-Av M.: Enumeration of coherent configurations of small orders (In preparation).
Acknowledgments
We are grateful for support from the Project Mobility—enhancing research, science and education at the Matej Bel University, ITMS code: 26110230082, under the Operational Program Education cofinanced by the European Social Fund. On different stages of this project we were using computer algebra package GAP. First version of the paper was reported by MK on the workshop “Peter Pal Palfy is 60”, Budapest, August 2015. We are much grateful to Sven Reichard for helpful cooperation and Gareth Jones and Misha Muzychuk for interesting comments. We thank two anonymous reviewers for their helpful critical comments. Finally, our thanks go to the editors of this collection, and especially to Jack Koolen for kind invitation and patience during the process of writing this text.
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In honor of Andries Brouwer’s 65th birthday.
Supported by the Mobility Grant, see details in Acknowledgements.
This is one of several papers published in Designs, Codes and Cryptography comprising the special issue in honor of Andries Brouwer’s 65th birthday.
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Klin, M., Ziv-Av, M. A non-Schurian coherent configuration on 14 points exists. Des. Codes Cryptogr. 84, 203–221 (2017). https://doi.org/10.1007/s10623-016-0258-8
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DOI: https://doi.org/10.1007/s10623-016-0258-8
Keywords
- Coherent configuration
- Association scheme
- Centralizer algebra
- Automorphism group
- Cayley graph
- Quaternion group