Selected Topics in AGT Lecture 4 Introduction to Schur Rings

Transcription

1 Selected Topics in AGT Lecture 4 Introduction to Schur Rings Mikhail Klin (BGU and UMB) September 14 18, 2015 M. Klin Selected topics in AGT September / 75

2 1 Schur rings as a particular case of association schemes 2 Cayley graphs 3 Circulant graphs 4 Isomorphism problem for circulant graphs 5 S-rings over cyclic groups 6 General idea of an S-ring 7 Non-Schurian S-rings 8 Primitive and imprimitive S-rings 9 Association schemes vs. S-rings M. Klin Selected topics in AGT September / 75

3 Recall that a regular permutation group is a transitive group of degree and order n. Each regular permutation group (H, H) appears via right regular action: with the element g H we associate permutation ĝ Sym(H) such that x ĝ = xg for all x X. Sometimes we denote (Ĥ, H). M. Klin Selected topics in AGT September / 75

4 Recall that for each association scheme M = (Ω, {R 0,..., R d }) we consider group Aut(M) of (combinatorial) automorphisms: Aut(M) = d Aut(Γ i ), i=0 Γ i = (Ω, R i ) is a basic graph of M, 0 i d. M. Klin Selected topics in AGT September / 75

5 Roughly speaking, a Schur ring is a special compact representation of an association scheme M, whose group Aut(M) contains a regular subgroup (H, H). Then we speak of a Schur ring (briefly S-ring) over H. Each basic graph appears as a suitable Cayley graph. M. Klin Selected topics in AGT September / 75

6 Example 4.1 (Example 1.3 revisited) Γ 0 Γ 1 Γ 2 Γ 3 This is metric scheme M of rank 3. Aut(M) = D 6, dihedral group of order 12 and degree 6. M. Klin Selected topics in AGT September / 75

7 D 6 contains regular cyclic group (Z 6, Z 6 ), acting on {0, 1, 2, 3, 4, 5}. Z 6 = g, g = (0, 1, 2, 3, 4, 5). We substitute each basic graph Γ i by so-called simple quantity, which consists of neighbors of vertex 0. γ = {0, 1, 5, 2, 4, 3}. M. Klin Selected topics in AGT September / 75

8 Our goals now are to understand the advantages of such a notion. In particular, we will learn more formally the concept of a Cayley graph. Also we have to figure out what are the links of S-rings with group rings. M. Klin Selected topics in AGT September / 75

9 1 Schur rings as a particular case of association schemes 2 Cayley graphs 3 Circulant graphs 4 Isomorphism problem for circulant graphs 5 S-rings over cyclic groups 6 General idea of an S-ring 7 Non-Schurian S-rings 8 Primitive and imprimitive S-rings 9 Association schemes vs. S-rings M. Klin Selected topics in AGT September / 75

10 Let H be a group (multiplicative notation), S H. We construct a Cayley graph Γ = (H, R) = Γ(X ) with the vertex set H and set R = {(x, yx) x H, y S}. Check that Aut(Γ) contains a regular subgroup (Ĥ, H). In this case S is usually called the connection set of Γ. M. Klin Selected topics in AGT September / 75

11 Example 4.2 H = S 3 = {g 1, g 2, g 3, g 4, g 5, g 6 } = {e, (1, 2), (1, 3), (2, 3), (1, 2, 3), (1, 3, 2)}. (Ĥ, H) = {ĝ 1, ĝ 2, ĝ 3, ĝ 4, ĝ 5, ĝ 6 }. E.g. ĝ 2 = (1, 2)(3, 5)(4, 6), ĝ 5 = (1, 5, 6)(2, 3, 4). Define S = {(1, 3), (1, 2, 3)}. 1 2 Γ(S) : M. Klin Selected topics in AGT September / 75

12 It turns out the Γ(S) is the smallest proper example of a directed strongly regular graph. Indeed, Γ(S) has v = 6 vertices, valency k = 2; each vertex is end point of t = 1 undirected edges; each arc is in λ = 0 triangles; each non-arc is in µ = 1 triangles. Parameter set: (6, 2, 1, 0, 1). M. Klin Selected topics in AGT September / 75

13 Example 4.1 (Revisited) Here we consider the cyclic group Z 6 in additive notation. All of our basic graphs are Cayley graphs over Z 6. Each connection set defines a Cayley graph over Z 6. We have a special partition of Z 6. M. Klin Selected topics in AGT September / 75

14 1 Schur rings as a particular case of association schemes 2 Cayley graphs 3 Circulant graphs 4 Isomorphism problem for circulant graphs 5 S-rings over cyclic groups 6 General idea of an S-ring 7 Non-Schurian S-rings 8 Primitive and imprimitive S-rings 9 Association schemes vs. S-rings M. Klin Selected topics in AGT September / 75

15 Z n = {0, 1,..., n 1} denotes the ring of residue classes modulo n, n N, with operations,. For X Z n, let Γ = Γ(Z n, X ) be the directed graph with the vertex set Z n and the arc set R = {(x, y) x, y Z n, x y X }. Here ( x) = ( 1) x. We call the graph Γ(Z n, X ) a circulant graph (or Cayley graph over Z n ), defined by the connection set X. M. Klin Selected topics in AGT September / 75

16 A main property of a circulant graph (briefly, circulant): a cyclic shift x x 1, is an automorphism. In our definition, a circulant graph is labeled graph: its vertices carry labels from Z n, cyclic shift is an automorphism of a graph. M. Klin Selected topics in AGT September / 75

17 Example 4.3 (A few abstract circulants) 2 K 3 Octahedron Möbius ladder M 4 M. Klin Selected topics in AGT September / 75

18 Example 4.3 (cont.) Representation of the above graphs as circulants: David star 3 K 2 = L(K 4 ) Γ(Z 6, {2, 4}) Γ(Z 6, {1, 2, 4, 5}) Γ(Z 8, {1, 4, 7} M. Klin Selected topics in AGT September / 75

19 Let Z n be the set of invertible elements of the ring Z n. Recall that Z n = φ(n), where φ denotes the Euler function. Let Aff (1, n) = {µ a,b a Z n, b Z n }, where µ a,b : Z n Z n is defined by µ a,b (x) = a x b. Clearly, Aff (1, n) is a permutation group, acting on Z n. Aff (1, n) is full affine group over Z n. M. Klin Selected topics in AGT September / 75

20 Proposition 4.1 Let Γ = Γ(Z n, X ) be a circulant graph and let µ Aff (1, n). Then: 1 The image Γ µ of Γ under the permutation µ is again a circulant. 2 Graphs Γ and Γ µ are isomorphic. Remark. Graphs Γ and Γ µ are said to be multiplicatively isomorphic if µ = µ a = µ a,0 for a Z n. M. Klin Selected topics in AGT September / 75

21 Example Γ 1 = Γ(Z 6, {2, 3}) Γ 1 = Γ(Z 6, {3, 4}) The graphs Γ 1 and Γ 2 are multiplicatively isomorphic with µ = µ 5. M. Klin Selected topics in AGT September / 75

22 1 Schur rings as a particular case of association schemes 2 Cayley graphs 3 Circulant graphs 4 Isomorphism problem for circulant graphs 5 S-rings over cyclic groups 6 General idea of an S-ring 7 Non-Schurian S-rings 8 Primitive and imprimitive S-rings 9 Association schemes vs. S-rings M. Klin Selected topics in AGT September / 75

23 Ádám s conjecture. In 1967 András Ádám posed the following question: Are any isomorphic circulant graphs multiplicatively isomorphic? The first counterexamples were quickly found for n = 8 (directed case) and for n = 16 (undirected case). To avoid more sophistication, let us consider counterexamples for n = 9. M. Klin Selected topics in AGT September / 75

24 Example 4.5 Let Γ 1 = Γ(Z 9, {1, 3, 4, 7}), Γ 2 = Γ(Z 9, {1, 4, 6, 7}). Γ 1 and Γ 2 are isomorphic, see picture Γ Γ 2 They are not multiplicatively isomorphic (simple inspection). M. Klin Selected topics in AGT September / 75

25 Finally, Ádám s conjecture was transformed into what is called Ádám s problem: Find necessary and sufficient conditions for two circulants to be isomorphic. Desired answer was to be formulated in terms similar to the notion of multiplicative isomorphism. The theory of S-rings was successfully used for the solution of this problem. M. Klin Selected topics in AGT September / 75

26 The first attempts to use S-rings were done by K. and Pöschel (1978). Full solution of this isomorphism problem was obtained by M. Muzychuk (2004). S. Evdokimov and I. Ponomarenko found efficient algorithms to recognize circulants (2003). Still this area of applications of S-rings is flourishing. M. Klin Selected topics in AGT September / 75

27 Picture of Ádám, K., Pöschel: Budapest, August M. Klin Selected topics in AGT September / 75

28 1 Schur rings as a particular case of association schemes 2 Cayley graphs 3 Circulant graphs 4 Isomorphism problem for circulant graphs 5 S-rings over cyclic groups 6 General idea of an S-ring 7 Non-Schurian S-rings 8 Primitive and imprimitive S-rings 9 Association schemes vs. S-rings M. Klin Selected topics in AGT September / 75

29 General idea: Start from the partition of Z n to X 0 = {0}, X 1,..., X d ; for simplicity, write X i as underlined list ; extend the operation to the sets X i ; check that we are getting S-ring. Let us learn everything on the level of example. M. Klin Selected topics in AGT September / 75

30 Example 4.6 Let n = 9, consider the simple quantities 0, 3, 6, 1, 4, 7, 2, 5, 8. Check that: 3 3 = 6, 3 6 = 0, 3 1, 4, 7 = 1, 4, 7,..., 1, 4, 7 1, 4, 7 = 2, 5, 8, 5, 8, 2, 8, 2, 5 = 3 2, 5, 8, 1, 4, 7 2, 5, 8 = ,... 3 ( 1) = 6 6 ( 1) = 3 1, 4, 7 ( 1) = 2, 5, 8. M. Klin Selected topics in AGT September / 75

31 Each time we get that addition in Z 9 of any two simple quantities is a linear combinations of simple quantities. Also, multiplication in Z 9 by ( 1) of a simple quantity is a simple quantity. We have to understand our example in more formal terms. M. Klin Selected topics in AGT September / 75

32 Namely, we are working in a group ring over Z 9. We are taking its Z-submodule. We require that this Z-submodule will have some extra properties. Then we reach S-ring over Z n. M. Klin Selected topics in AGT September / 75

33 1 Schur rings as a particular case of association schemes 2 Cayley graphs 3 Circulant graphs 4 Isomorphism problem for circulant graphs 5 S-rings over cyclic groups 6 General idea of an S-ring 7 Non-Schurian S-rings 8 Primitive and imprimitive S-rings 9 Association schemes vs. S-rings M. Klin Selected topics in AGT September / 75

34 Let (G, A) be a transitive permutation group having a regular subgroup (A, A). Let H = G 0 be the stabilizer in G of the neutral element 0 A, and let X 0 = {0}, X 1,..., X d be the set of orbits of permutation group (H, A). Finally, let { d } M = M(G, A) = λ i X i λ i Z, i = 0, 1,..., d i=0 be the Z-module with basis X 0, X 1,..., X d, where X i = x X i x. M. Klin Selected topics in AGT September / 75

35 In this case, we call M the transitivity module of the permutation group (G, A) (with regular subgroup (A, A)). M. Klin Selected topics in AGT September / 75

36 Theorem 4.2 (Schur-Wielandt) With previous notation let M = M(G, A) be the transitivity module of the transitive permutation group (G, A) with regular subgroup A. Then i M is a subring of Z[A]; ii for all i {0, 1,..., d}, Xi t = X i for suitable i {0, 1,..., d}. (Here X t is defined as X t = { x x X }.) An outline of a proof is available in G. Jones notes (2014). M. Klin Selected topics in AGT September / 75

37 The proof of the theorem actually establishes an isomorphism between the centralizer ring V (G, A) and the subring M of the integral group ring Z[A]. Thus all computations in V (G, A) can be effectively carried out in the transitivity module M(G, A) of (G, A). The axiomatic development introduced below goes back to the classical paper of I. Schur (1933). M. Klin Selected topics in AGT September / 75

38 A subring S Z[A] is called an S-ring over group A if the following conditions are satisfied: (S1) S has a basis consisting of T 0, T 1,..., T d ; (S2) T i T j = for all i j; (S3) T 0 = {0}, d i=0 T i = A; (S4) for each i {0, 1,..., d}, Ti t i {0, 1,..., d}. = T i for suitable Here T t i is set of inverse elements to the elements in T i. M. Klin Selected topics in AGT September / 75

39 We often refer to T 0, T 1,..., T d as basis quantities, and to T 0, T 1,..., T d as basis sets of the corresponding S-ring S. In this situation, we also write S = T 0, T 1,..., T d, and we say S has rank d + 1 and order A. M. Klin Selected topics in AGT September / 75

40 It follows that for every transitive permutation group (G, A) having a regular subgroup (A, A), its transitivity module M(G, A) is an S-ring over A. We shall call such S-rings Schurian (or of Schur-type). The perception is that I. Schur himself believed that every S-ring is a transitivity module of a suitable permutation group. M. Klin Selected topics in AGT September / 75

41 First counterexamples to Schur s informal conjecture were obtained at the time of life of I. Schur. They appear over elementary abelian group Z 2 p, p an odd prime. Nice presentation was given at the seminal book of H. Wielandt (1964). We will follow it. M. Klin Selected topics in AGT September / 75

42 1 Schur rings as a particular case of association schemes 2 Cayley graphs 3 Circulant graphs 4 Isomorphism problem for circulant graphs 5 S-rings over cyclic groups 6 General idea of an S-ring 7 Non-Schurian S-rings 8 Primitive and imprimitive S-rings 9 Association schemes vs. S-rings M. Klin Selected topics in AGT September / 75

43 Non-Schurian examples for a wider class of association schemes and coherent configurations were discussed already in lecture 3 by MZA. Here we restrict ourselves to S-rings. We start from the classical example exposed by Wielandt. M. Klin Selected topics in AGT September / 75

44 Example 4.7 (H. Wielandt) Let A be the elementary abelian group E 25, and let S = T 0, T 1, T 2, T 3, T 4, where T 0 = {00}, T 1 = {01, 02, 03, 04}, T 2 = {10, 20, 30, 40}, T 3 = {11, 22, 33, 44}, T 4 = A \ (T 0 T 1 T 2 T 3 ). It is easy to check that S is an S-ring. This also follows from the consideration of so-called amorphic S-rings. M. Klin Selected topics in AGT September / 75

45 Let us show that S is a non-schurian S-ring. Assume that S is Schurian and consider the complete color graph Γ on 25 vertices associated to S. (That is, Γ is the complete directed graph with vertex set E 25 wherein arc (g, h) receives color i if and only if h g T i.) The two edges δ = {00, 12} and ɛ = {00, 13} both have color 4 in Γ since 12, 13 T 4. M. Klin Selected topics in AGT September / 75

46 Consider 4-vertex subgraph of Γ at the left containing edge δ X 2 XX Assume that a suitable element of Aut( Γ) sends it to the right. Clearly all colors are preserved. Thus, the edge δ maps to ɛ. M. Klin Selected topics in AGT September / 75

47 We have just four possibilities for the value of X. Inspecting the edge {13, XX } of color 2 together with the edge {0X, 13} of color 3, we get a contradiction between X = 2 and absence of solution in the last case. M. Klin Selected topics in AGT September / 75

48 A Latin square Σ of order n is an n n array (n rows and n columns) in which n copies of the numbers 0, 1, 2,..., n 1 collectively appear but without repetition of any number in a single row or column. A Latin square of order 4 is depicted in below M. Klin Selected topics in AGT September / 75

49 Σ cells Let us construct the graph Γ = Γ(Σ) of this Latin square: Vertices of the graph are the 16 cells of the square, labeled sequentially as in the table on the right. An edge is drawn between any two vertices which are either in the same row or column, or which are occupied by the same number in Σ. M. Klin Selected topics in AGT September / 75

50 Thus in our example vertex 0 is connected by an edge to each of the vertices 1, 2, 3, 4, 7, 8, 10, 12, 13. The adjacency matrix of Γ can be easily written, though the actual graph looks somewhat complicated. M. Klin Selected topics in AGT September / 75

51 In order to get a nicer pictorial representation we consider instead the complementary graph Γ, the so-called Shrikhande graph, which we denote by Sh. Sh has edges where Γ has non-edges and vice versa, e.g., vertex 0 is connected to vertices 5, 6, 9, 11, 14, 15. Sh is a non-planar graph, but it has a toroidal embedding. M. Klin Selected topics in AGT September / 75

52 The torus diagram of Sh shown below M. Klin Selected topics in AGT September / 75

53 Adjacency matrix of Sh A = M. Klin Selected topics in AGT September / 75

54 Because the graph Γ = Sh is obtained from the multiplication table of Z 4, it is easy to check that Aut(Γ) contains regular subgroup Z 2 4. In fact, Aut(Γ) has order 192. Thus, we are working inside of an S-ring M. M. Klin Selected topics in AGT September / 75

55 This S-ring M has 3 basic quantities T 0, T 1, T 2, where T 1 = 6, T 2 = 9. An exercise for an interested participant is to get a multiplication table for M. M. Klin Selected topics in AGT September / 75

56 We wish to show that M is non-schurian S-ring. For this purpose it is convenient to work with the graph Γ of valency 9. Again we will manipulate with a suitable 4-vertex configuration in Γ. M. Klin Selected topics in AGT September / 75

57 A useful characteristic of an edge is the number α of 4-vertex complete subgraphs (4-cliques) in which this edge is involved. Let us determine α for each of the edges {0, 1} and {0, 10} of Γ. These numbers are easily read off from the figures (a) and (b), respectively. (The labeling scheme for vertices is the same.) M. Klin Selected topics in AGT September / 75

58 A 4-clique involving edge {0, 1} is found whenever two common neighbors of 0 and 1 are connected by an edge. Common neighbors of 0 and 1 are 2, 3, 4, 13, and of these only 2 and 3 are connected by an edge. Hence α({0, 1}) = 1. Common neighbors of 0 and 10 are 2, 7, 8, 13, and of these both 2 and 8 and 7 and 13 are connected by edges, which gives α({0, 10}) = 2. M. Klin Selected topics in AGT September / 75

59 (a) (b) M. Klin Selected topics in AGT September / 75

60 Thus we again are able to distinguish edges of a basic graph by counting some local invariants. This basic graph is not arc-transitive. In other words, our S-ring of rank 3 is non-schurian. M. Klin Selected topics in AGT September / 75

61 The technique we used is called the 4-vertex condition. Its pros and cons are an issue in AGT. It is of special interest for rank 3 association schemes. M. Klin Selected topics in AGT September / 75

62 1 Schur rings as a particular case of association schemes 2 Cayley graphs 3 Circulant graphs 4 Isomorphism problem for circulant graphs 5 S-rings over cyclic groups 6 General idea of an S-ring 7 Non-Schurian S-rings 8 Primitive and imprimitive S-rings 9 Association schemes vs. S-rings M. Klin Selected topics in AGT September / 75

63 Let S be an S-ring over a group A, and let U be a subgroup of A. We say that U belongs to S if U S. We call the S-ring S primitive if 0 and A are the only subgroups of A which belong to S. Otherwise, we call S imprimitive. M. Klin Selected topics in AGT September / 75

64 The following proposition is evident. Proposition 4.3 (i) For p prime, every S-ring over Z p is primitive; (ii) The group ring Z[A] is primitive if and only if A = Z p, p prime. M. Klin Selected topics in AGT September / 75

65 The concept of primitivity of an S-ring played a significant role in the original paper by I. Schur (1933). Below we just briefly mention main ideas. See also lecture notes by G. Jones (2014) on the site. M. Klin Selected topics in AGT September / 75

66 Let A be a finite group. We call A a Burnside group (or B-group) if every overgroup (G, A) in S A of the right regular representation (A, A) is either an imprimitive or doubly transitive permutation group. This notion goes back to W. Burnside, who proved that every cyclic group of proper prime power order is a B-group. M. Klin Selected topics in AGT September / 75

67 Proposition 4.4 (Schur) Let A be a group with the property that every S-ring over A of rank at least 3 is imprimitive. Then A is a B-group. Proof. We consider the transitivity module S = M(G, A), where (G, A) is any overgroup of (A, A) in S A. If (G, A) is 2-transitive then S has rank 2, otherwise S has rank at least 3 so is imprimitive by assumption. But this implies that V (G, A) is imprimitive, whence (G, A) is an imprimitive permutation group. M. Klin Selected topics in AGT September / 75

68 Using these concepts, Schur proved in his seminal paper that a cyclic group Z n of composite order n is a B-group. Oppositely to Burnside, his proof does not require use of character theory. Extra significant concept: S-ring of traces, a rational S-ring in modern terms. M. Klin Selected topics in AGT September / 75

69 1 Schur rings as a particular case of association schemes 2 Cayley graphs 3 Circulant graphs 4 Isomorphism problem for circulant graphs 5 S-rings over cyclic groups 6 General idea of an S-ring 7 Non-Schurian S-rings 8 Primitive and imprimitive S-rings 9 Association schemes vs. S-rings M. Klin Selected topics in AGT September / 75

70 Up to order 9, each association scheme has at least one representation (up to isomorphism) as a suitable S-ring. For order 10 this is not true. The famous Petersen graph P provides a counterexample. M. Klin Selected topics in AGT September / 75

71 Outline of a proof: P = L(K 5 ); Aut(P) = S 5 ; No subgroup of S 5 of order 10 acts transitively on the vertices of P. See also lecture notes by G. Jones. M. Klin Selected topics in AGT September / 75

72 An attempt of generalization. Marušič-Klin conjecture: If M is an association scheme with transitive G = Aut(M), then G contains a fixed-point free element of a suitable prime order. M. Klin Selected topics in AGT September / 75

73 The conjecture is true for every S-ring. It is also true for the Petersen graph. Many attempts of investigations. Still there is no counterexample. M. Klin Selected topics in AGT September / 75

74 Some references P. J. Cameron, M. Giudici, G. A. Jones, W. M. Kantor, M. H. Klin, D. Marušič, L. A. Nowitz, Transitive permutation groups without semiregular subgroups, J. London Math. Soc. (2) 66 (2002), no. 2, M. Klin, I. Kovács, Automorphism groups of rational circulant graphs. Electron. J. Combin. 19 (2012), no. 1, Paper 35, 52 pp. M. Muzychuk, M. Klin, R. Pöschel, The isomorphism problem for circulant graphs via Schur ring theory, in: Codes and association schemes, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 56 (Amer. Math. Soc., Providence, RI, 2001) M. Muzychuk, A solution of the isomorphism problem for circulant graphs. Proc. London Math. Soc. (3) 88 (2004), no. 1, I. Schur, Zur Theorie der einfach transitiven Permutationsgruppen, S. B. Preuss. Akad. Wiss., Phys.-Math. Kl, 1933, A. E. Brouwer, Shrikhande graph, H. Wielandt, Finite permutation groups. Acad. Press, M. Klin Selected topics in AGT September / 75

75 Thank you! M. Klin Selected topics in AGT September / 75