Coherent Configurations

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1 Coherent Configurations Lecture 10: Miscellanea: Some research tasks in AGT Mikhail Klin (Ben-Gurion University) September 1 5, 2014 M. Klin (BGU) Miscellanea 1 / 67

2 Preamble We will discuss a number of research tasks of different level and significance. Most of them are formulated in terms of color graphs, typically CCs or even ASs. M. Klin (BGU) Miscellanea 2 / 67

3 Preamble As a rule, the tasks are related to computations: to perform some computations; to explain (interpret) computer results; to provide theoretical generalizations of computer aided activities. M. Klin (BGU) Miscellanea 3 / 67

4 Preamble A list of related publications will be provided at the end of lecture. Participants are welcome to communicate with MK, regarding extra clarifications and comments (personally or via ). M. Klin (BGU) Miscellanea 4 / 67

5 Part A. Nice interpretation is requested As a rule, an interpretation is expected, though helpful explanation might serve as a suitable palliative. M. Klin (BGU) Miscellanea 5 / 67

6 A.1: SRGs on 25 and 26 vertices Up to isomorphism, there exist 15 SRGs with parameters (25, 12, 5, 6) and 10 SRGs for (26, 10, 3, 4). First case, when the use of computers for full classification is essential. Also first time, when graphs may have trivial automorphism group (of order 1). M. Klin (BGU) Miscellanea 6 / 67

7 A.1: SRGs on 25 and 26 vertices The results were obtained independently by a few groups ( ). There are four remarkable geometric graphs, which are coming from Latin squares of order 5 and Steiner triple systems on 13 points. M. Klin (BGU) Miscellanea 7 / 67

8 A.1: SRGs on 25 and 26 vertices All other graphs are obtained from geometrical ones, using procedure of switching (in the sense of J.J. Seidel). Two-graphs are playing essential role in classification. M. Klin (BGU) Miscellanea 8 / 67

9 A.1: SRGs on 25 and 26 vertices Long-standing challenge: Repeate full classification without the (essential) use of a computer. Our first attempts did not succeed: new ideas are requested! M. Klin (BGU) Miscellanea 9 / 67

10 A.2: Partial geometries pg(5,7,3) We use notation, traditional in design theory (due to R.C.Bose), oppositely to the so-called Belgian notation. This is the smallest case when classification of partial geometries is becoming non-trivial. M. Klin (BGU) Miscellanea 10 / 67

11 A.2: Partial geometries pg(5,7,3) Up to isomorphism there exist two pg(5,7,3), they have 45 points and 63 blocks. The result belongs to R.Mathon (1981) via exhaustive computer search. The automorphism groups have orders 1512 and 216. M. Klin (BGU) Miscellanea 11 / 67

12 A.2: Partial geometries pg(5,7,3) MK, SR and A.Woldar obtained nice interpretation of both geometries with the aid of so-called overlarge sets of affine designs with 8 points. This is, indeed, an interpretation. Prove without the use of a computer, that there are just two geometries. M. Klin (BGU) Miscellanea 12 / 67

13 A.3: An SRG on 1716 vertices There is a number of SRGs, which appear via mergings of classes of Johnson association scheme. All were discovered by MK (1974) via the use of a computer. Some are Schurian, some not. M. Klin (BGU) Miscellanea 13 / 67

14 A.3: An SRG on 1716 vertices The largest one Γ has parameters (1716, 833, 400, 306), Aut(Γ) = S 13. No nice computer free interpretation of appearance of Γ has been done. Probably, pseudo M 13 puzzle (John Conway) might be of help (aka pseudogroup extension of Mathieu group M 12 ). M. Klin (BGU) Miscellanea 14 / 67

15 A.3: An SRG on 1716 vertices Conjecture (MK) All mergings of classes of Johnson schemes are known. Partial confirmation by Misha Muzychuk and Vasyl Ustimenko (theory and computer). Extremely difficult to get more progress. M. Klin (BGU) Miscellanea 15 / 67

16 Part B: Partial results are known Here we consider a few problems, where first success was achieved, starting with a computer, and using also theoretical arguments. Next steps are welcome, again via amalgamation of computation and theory. M. Klin (BGU) Miscellanea 16 / 67

17 B.1: Partial geometries pg(8,9,4) Partial geometries pg(8,9,4) have 120 points and 135 blocks. Both point and block graphs have classical parameters. M. Klin (BGU) Miscellanea 17 / 67

18 B.1: Partial geometries pg(8,9,4) An example of a partial geometry pg(8,9,4) was known for a long time, though in a few different incarnations: F.De Clerck, R.H. Dye, J.A. Thas (1980); A.M.Cohen (1981); W.Haemers, J.H. van Lint (1982). M. Klin (BGU) Miscellanea 18 / 67

19 B.1: Partial geometries pg(8,9,4) Relying on ideas of W.M. Kantor and using a computer, V.D. Tonchev (1984) proved that all constructions provide the same (up to isomorphism) incidence structure. M. Klin (BGU) Miscellanea 19 / 67

20 B.1: Partial geometries pg(8,9,4) For a while it was suspected that other examples do not exist. New splash of activities around 1997: MK and SR; R. Mathon and A.P. Street. M. Klin (BGU) Miscellanea 20 / 67

21 B.1: Partial geometries pg(8,9,4) 7 more geometries were discovered (computer), four of them have the same point graphs. This common point graph was known before (A. Brouwer, A.V. Ivanov, MK). M. Klin (BGU) Miscellanea 21 / 67

22 B.1: Partial geometries pg(8,9,4) Thus, altogether there are known 8 geometries with different point graphs (one of them is rank 3 graph). Are there more pg(8,9,4)? M. Klin (BGU) Miscellanea 22 / 67

23 B.2: Highly symmetrical proper loops To every quasigroup (and particular loop) Q of order n we associate an SRG Γ(Q) with the parameters (n 2, 3(n 1), n, 6). We say that loops Q 1 and Q 2 belong to different main classes if Γ(Q 1 ) and Γ(Q 2 ) are not isomorphic. M. Klin (BGU) Miscellanea 23 / 67

24 B.2: Highly symmetrical proper loops Numbers of main classes for small n: n # Nice loop (or Latin square) Q Γ = Γ(Q) has a rich group Aut(Γ). M. Klin (BGU) Miscellanea 24 / 67

25 B.2: Highly symmetrical proper loops Classical result, due to E. Schönhardt (1930): If Q is a group, then Aut(Γ(Q)) is transitive and its structure is (Q 2 : Aut(Q)).S 3. M. Klin (BGU) Miscellanea 25 / 67

26 B.2: Highly symmetrical proper loops Moreover, in this case G = Aut(Γ(Q)) contains a regular subgroup (of order n 2 ). In a more specific case we speak about highly symmetric loop (consider direction preserving subgroup of G). M. Klin (BGU) Miscellanea 26 / 67

27 B.2: Highly symmetrical proper loops Each group is a highly symmetric loop. Highly symmetric loop, which does not belong to any main class of a group, is called proper. Nice concept on the edge between AGT and algebra. M. Klin (BGU) Miscellanea 27 / 67

28 B.2: Highly symmetrical proper loops No proper hs-loops for orders up to 5. Just one such loop for order n = 6. Really reasonably rare structures. M. Klin (BGU) Miscellanea 28 / 67

29 B.2: Highly symmetrical proper loops A.Heinze and MK: investigation of infinite series of proper hs-loops of order 2p, p a prime. MK, N.Kriger and A. Woldar get full classification of such loops of order 2p: just one class, so-called Wilson loops. M. Klin (BGU) Miscellanea 29 / 67

30 B.2: Highly symmetrical proper loops Next interesting case is n = 3p, or, more generally, n = pq, p and q distinct odd primes. Repeat similar classification. M. Klin (BGU) Miscellanea 30 / 67

31 B.3: Elusive groups and polycirculant conjecture A permutation g e is called semiregular, if all its cycles have the same length (at least 2). Most of transitive groups contain a semiregular permutation. M. Klin (BGU) Miscellanea 31 / 67

32 B.3: Elusive groups and polycirculant conjecture For example, automorphism group of Petersen graph contains an automorphism of order 5 (two cycles of length 5). Compact depiction of the famous diagram: 1 2 l = 5 M. Klin (BGU) Miscellanea 32 / 67

33 B.3: Elusive groups and polycirculant conjecture A transitive permutation group G of degree n is called elusive if it does not contain a non-identity semiregular permutation. Smallest degree of elusive group is n = 12: Famous M 11 in transitive action of degree 12. M. Klin (BGU) Miscellanea 33 / 67

34 B.3: Elusive groups and polycirculant conjecture They are indeed elusive: very restricted classes of examples. Recall that a permutation group (G, Ω) is called 2-closed, if G = Aut(Γ) for a suitable color graph with vertex set Ω. M. Klin (BGU) Miscellanea 34 / 67

35 B.3: Elusive groups and polycirculant conjecture I. Faradžev, MK, O. Korsunskaya (1988) inspected, using a computer, many examples of 2-closed permutation groups. Each time we found a semiregular automorphism and depicted compact diagram. M. Klin (BGU) Miscellanea 35 / 67

36 B.3: Elusive groups and polycirculant conjecture Conjecture (MK, generalizing Dragan Marušič): Any 2-closed permutation group is not elusive. It is usually called the polycirculant conjecture. M. Klin (BGU) Miscellanea 36 / 67

37 B.3: Elusive groups and polycirculant conjecture A lot of theoretical results confirm this conjecture for many particular cases. Typically, CFSG is used. See recent surveys by Michael Giudici. M. Klin (BGU) Miscellanea 37 / 67

38 B.3: Elusive groups and polycirculant conjecture Why not to try to confirm this conjecture, exploiting more essentially graph-theoretical techniques? New ideas are requested. M. Klin (BGU) Miscellanea 38 / 67

39 Part C: Open problems Most problems below rely on some concrete objects in AGT with prescribed parameters. A few times we simply consider the smallest members of some infinite series of parameter sets. M. Klin (BGU) Miscellanea 39 / 67

40 C.1: Packing of 7 copies of HoSi in K 50 Recall that the Hoffman-Singleton graph is an SRG(50,7,0,1) = 7 7. We can think of color graph: each of 7 graphs is isomorphic to HoSi. M. Klin (BGU) Miscellanea 40 / 67

41 C.1: Packing of 7 copies of HoSi in K 50 M. Meszka and J. Šiagiová found 5 disjoint copies of HoSi, using favourite voltage graph methods. Prove that packing with 7 copies is impossible. Remark: Corresponding negative result for Petersen graph is known. M. Klin (BGU) Miscellanea 41 / 67

42 C.2: Symmetric association schemes with 3 classes Edwin van Dam developed elements of the theory of 3-class association schemes. He also did a lot of enumeration. The results are presented in helpful tables. M. Klin (BGU) Miscellanea 42 / 67

43 C.2: Symmetric association schemes with 3 classes New parameter sets (up to 100 vertices) remain open for about 15 years (no examples, no killing). M. Klin (BGU) Miscellanea 43 / 67

44 C.2: Symmetric association schemes with 3 classes A few open cases (small sample): n valencies spectrum 52 17,17, irrational 55 18,18,18 18, irrational 60 21,14,24 21, 3 32, 4 24, ,40,5 20, 2 44, 2 10, ,45,20 30, 6 30, 2 45, 6 20 M. Klin (BGU) Miscellanea 44 / 67

45 C.3: Pair of dual SRGs on 96 vertices Consider open parameter set of SRGs on 96 vertices: n = 96, k = 45, l = 50, λ = 21, µ = 18 with spectrum 45, 9 20, Assume in addition, that there exists a spectrally-dual SRG on 96 vertices (with respect to a suitable commutative fission). M. Klin (BGU) Miscellanea 45 / 67

46 C.3: Pair of dual SRGs on 96 vertices Prove or disprove existence of such a pair. Remark: In fact, this is one of the smallest possible cases for a new kind of duality in comparison with known ones. M. Klin (BGU) Miscellanea 46 / 67

47 C.4: SRG on 100 vertices The only open parameter set for SRGs on strictly 100 vertices: n = 100, k = 33, λ = 8, µ = 12. According to van Dam, such a graph might be embedded into 3-class scheme with valencies 22,33,44 (two different parameter sets). M. Klin (BGU) Miscellanea 47 / 67

48 C.5: tfsrg on 162 vertices The parameter set with n = 162, k = 21, l = 140, λ = 0, µ = 3 is the smallest open case for a possible primitive tfsrg. Covering of this case is an ambitious, though not absolutely hopeless research task. M. Klin (BGU) Miscellanea 48 / 67

49 C.6: Non-Schurian example of a 4-isoregular graph Let Γ = (V, E) be a graph. Valency val(s) of a subset S V is the amount of vertices in V, which are neighbours of each x S. Graph Γ is called t-isoregular if for each S, S t, val(s) depends only in the isomorphism type of the subgraph, induced by S. M. Klin (BGU) Miscellanea 49 / 67

50 C.6: Non-Schurian example of a 4-isoregular graph 1-isoregular graph = regular. 2-isoregular = strongly regular. Remark: this definition was suggested by Ja.Ju.Gol fand (born 1947), killed in Moscow at the end of 1990th. M. Klin (BGU) Miscellanea 50 / 67

51 C.6: Non-Schurian example of a 4-isoregular graph Absolutely regular graph is V -isoregular. Theorem (MK) All absolutely regular graphs are expired (up to complement) by n K m (imprimitive SRGs) and the graph L 2 (3). M. Klin (BGU) Miscellanea 51 / 67

52 C.6: Non-Schurian example of a 4-isoregular graph Figure : Γ = L 2 (3). Note that Γ = Γ. M. Klin (BGU) Miscellanea 52 / 67

53 C.6: Non-Schurian example of a 4-isoregular graph Theorem 10.1 [Gol fand, P.J.Cameron] Each 5-isoregular graph is absolutely regular. Remark. Cameron is using different terminology. M. Klin (BGU) Miscellanea 53 / 67

54 Theorem 10.2: [Gol fand, unpublished] Parameters of any proper 4-isoregular graph belong to the following series M(r): v = (2r + 1)(2r 2 (2r + 3) 1), k = 2r 3 (2r + 3), l = 2(r + 1) 3 (2r 1), λ = r(2r 1)(r 2 + r 1), µ = r 3 (2r + 3). M. Klin (BGU) Miscellanea 54 / 67

55 C.6: Non-Schurian example of a 4-isoregular graph Graph M(1) with parameters (27, 10, 16, 1, 5) exists and is unique. It is the famous Schläfli graph. In certain strict sense the most beautiful non-trivial graph. M. Klin (BGU) Miscellanea 55 / 67

56 C.6: Non-Schurian example of a 4-isoregular graph Graph M(2) with parameters (275, 112, 162, 30, 56) also exists and is unique. It is famous McLaughlin graph, related to a sporadic simple group. M. Klin (BGU) Miscellanea 56 / 67

57 C.6: Non-Schurian example of a 4-isoregular graph Graph M(3) with the parameters (1127, 486, 640, 165, 243) does not exist (easily follows from a result of A. Makhnev, 2002). The case M(4) with parameters (3159, 1408, 532, 704) was also considered by Makhnev (2011). Some partial negative results were obtained. M. Klin (BGU) Miscellanea 57 / 67

58 C.6: Non-Schurian example of a 4-isoregular graph A.Munemasa and B.Venkov obtained (in absolutely different context of tight spherical designs) results, which eliminate existence of M(r) for infinitely many values of r. Not visible explicitely for a non-perplexed reader. M. Klin (BGU) Miscellanea 58 / 67

59 C.6: Non-Schurian example of a 4-isoregular graph Conjecture. Graphs M(r) do not exist for r 3. A very ambitious problem! M. Klin (BGU) Miscellanea 59 / 67

60 Acknowledgements Thanks again to Štefan Gyürki, Danny Kalmanovich, Christian Pech, Sven Reichard, Matan Ziv-Av for crucial assistance in the preparation of the lectures. M. Klin (BGU) Miscellanea 60 / 67

61 References Brouwer, A.E.; Haemers, W.H.: Spectra of graphs. Universitext. Springer, New York, xiv+250 pp. ISBN: Cameron, P.J.; Giudici, M.; Jones, G.A.; Kantor, W.M.; Klin, M. H.; Marušič, D.; Nowitz, L.A.: Transitive permutation groups without semiregular subgroups. J. London Math. Soc. (2) 66 (2002), no. 2, M. Klin (BGU) Miscellanea 61 / 67

62 References De Clerck, F.: Partial and semipartial geometries: an update. Combinatorics 2000 (Gaeta). Discrete Math. 267 (2003), no. 1 3, van Dam, E.R.: Three-class association schemes. J. Algebraic Combin. 10 (1999), no. 1, M. Klin (BGU) Miscellanea 62 / 67

63 References Faradžev, I. A.; Klin, M. H.; Muzichuk, M. E.: Cellular rings and groups of automorphisms of graphs. Investigations in algebraic theory of combinatorial objects, 1 152, Math. Appl. (Soviet Ser.), 84, Kluwer Acad. Publ., Dordrecht, M. Klin (BGU) Miscellanea 63 / 67

64 References Klin, M.; Muzychuk, M.; Ziv-Av, M.: Higmanian rank-5 association schemes on 40 points. Michigan Math. J. 58 (2009), no. 1, Klin, M.; Kriger, N.; Woldar, A.: Classification of highly symmetrical translation loops of order 2p, p prime. Beitr. Algebra Geom. 55 (2014), no. 1, M. Klin (BGU) Miscellanea 64 / 67

65 References Klin, M. Ch.; Pöschel, R.; Rosenbaum, K.: Angewandte Algebra für Mathematiker und Informatiker. (German) [Applied algebra for mathematicians and information scientists] Einführung in gruppentheoretisch-kombinatorische Methoden. [Introduction to group-theoretical combinatorial methods] Friedr. Vieweg & Sohn, Braunschweig, pp. ISBN: M. Klin (BGU) Miscellanea 65 / 67

66 References Klin, M.; Pech, C.; Reichard, S.; Woldar, A.; Ziv-Av, M.: Examples of computer experimentation in algebraic combinatorics. Ars Math. Contemp. 3 (2010), no. 2, Šiagiová, J.; Meszka, M.: A covering construction for packing disjoint copies of the Hoffman-Singleton graph into K 50. J. Combin. Des. 11 (2003), no. 6, M. Klin (BGU) Miscellanea 66 / 67

67 Acknowledgements Thank you very much for interest and attention to this series of lectures! M. Klin (BGU) Miscellanea 67 / 67

Permutation groups, derangements and prime order elements

Permutation groups, derangements and prime order elements Tim Burness University of Southampton Isaac Newton Institute, Cambridge April 21, 2009 Overview 1. Introduction 2. Counting derangements: Jordan