J , the all- 1 matrix. (The corresponding only G -invariant - PDF document

✞ ✟ ✞ ✞ ☞ ☞ ✞ ✟ ✞ ✞ Association schemes and Permutation groups permutation groups Let G be a permutation group on Ω . Then the ✆ Ω characteristic functions of the orbits of G on Ω Peter J Cameron ✍✏✎ G form a coherent configuration ✑ . School of Mathematical Sciences Queen Mary, University of London ✍✏✎ G ✑ is an association scheme if and only if G is London E1 4NS, U.K. generously transitive , that is, any two points of Ω are p.j.cameron@qmw.ac.uk interchanged by some element of G . Durham, July 2001 When is there a non-trivial G -invariant association scheme? Joint work with P . P . Alejandro and R. A. Bailey 1 3 Definitions ✆ Ω zero-one matrices. Such �✂✁✂✁✄✁✂� A r A 0 ☎ 1 are Ω ✆ Ω . matrices represent subsets of Ω More definitions Coherent configuration: The transitive permutation group G is AS-free if the ☎ 1 r ∑ J , the all- 1 matrix. (The corresponding only G -invariant association scheme is the trivial A i (a) ☛ I ✝ 0 � J i I ✆ Ω .) scheme ✌ . subsets form a partition of Ω The transitive permutation group G is AS-friendly if s ☎ 1 ∑ A i I . (The diagonal is a union of classes.) (b) there is a unique minimal G -invariant association ✝ 0 i scheme. ☛ 0 �✂✁✂✁✂✁✂� r 1 (c) A A i ✠ , where ✡ is an involution on ✌ . i The transitive permutation group G is stratifiable if we r ☎ 1 ✍✏✎ G obtain an association scheme by symetrising ✑ , ∑ p k (d) A i A j i j A k . (The matrices span an algebra.) that is, adding A i to A i for non-symmetric A i . ✝ 0 k Association scheme: ✡ is the identity, i.e. all the relations are symmetric. (This implies that s 1 , that is, A 0 I : the diagonal is a single class.) 2 4

✌ ✌ � ✌ � ✔ ✌ � ☛ ✌ � � ✞ ✌ � ✌ � ✌ ✁ ✞ ✗ ✞ ✌ � ✌ � � ✌ ✒ � ✓ ✓ ✌ ✒ ✚ ✓ ✓ � ✒ ✌ ✓ ✓ ✞ ✒ � ✌ ✞ Regular groups Implications For a regular permutation group G , the following are The following implications hold between these equivalent: conditions and others from permutation group theory. 2 -transitive gen. trans. (a) G is AS-friendly; 2 -homogeneous stratifiable (b) G is stratifiable; AS-free AS-friendly primitive transitive ✆ A , where A is an (c) either G is abelian, or G Q 8 elementary abelian 2 -group. No implications reverse, and no more implications hold except possibly from “primitive” to “AS-friendly”. Sketch proof: (c) implies (b) by character theory; (b) All conditions in the table are closed under taking implies (a) trivial; (a) implies (c) by an ad hoc supergroups. argument using Dedekind’s Theorem. 5 7 An example AS-free groups We describe a partition of G 2 invariant under right An AS-free group is primitive, and is 2 -homogeneous, multiplication by giving a partition of G : to C G ✕ C almost simple, or of diagonal type. ☎ 1 � h ✑ : gh ✎ g corresponds ✌ . We have a coherent ☛ 1 configuration if and only if ✌ is a class and the class For, if imprimitive, it preserves the “group-divisible” sums span a subring of the group ring (a Schur ring ). association scheme; and, of the types in the O’Nan–Scott Theorem, groups of affine type are We get an association scheme if and only if each stratifiable, and groups of product type preserve class is inverse-closed. Hamming schemes. Let G be the dihedral group ☛ 1 ✖ a Obviously any 2 -homogeneous group is AS-free. � b : a 3 b 2 ✑ 2 � a 2 � b 2 1 � a � b � b ✎ ab ☛✘☛ 1 ☛ a ☛ b ☛ ab Examples are known of almost simple AS-free � a 2 � a 2 b Then ✌✙✌ gives an groups which are not 2 -transitive, but they are quite ✆ 2 rectangle. association scheme, the 3 hard to find. The smallest known example has ☛✘☛ 1 degree 234 . ☛ a ☛ ab ☛ a 2 b � a 2 � b Similarly for ✌✘✌ and ☛✘☛ 1 ☛ a ☛ a 2 b ☛ b � a 2 � ab ✌✙✌ . No examples of AS-free groups of diagonal type are ☛✙☛ 1 ☛ a ☛ b ☛ ab ☛ a 2 b � a 2 known. Any such group must have at least four But ✌✙✌ does not give an simple factors in its socle. association scheme. So this group is not AS-friendly. 6 8

✞ ✕ ✧ ✞ ✞ ✚ ✣ ✣ ✧ ✑ ✣ ✣ ✞ Diagonal groups General diagonal groups Let T be a group and n a positive integer. Then We proved the following theorem: � n D ✎ T ✑ is the permutation group on the set ☛✜✛ t 1 �✂✁✂✁✂✁✄� t n ✢ : t 1 �✂✁✂✁✂✁✂� t n Ω T ✌ generated by � n (a) If T is abelian then D ✎ T ✑ is generously permutations of the following types: transitive. right translations by T n ; � n 8 , then (b) If D ✎ T ✑ is generously transitive with n T is abelian. automorphisms of T (acting in the same way on � 7 (c) If D ✎ T ✑ is generously transitive, then either T is each coordinate); abelian, or T Q 8 . permutations of the coordinates; � n 9 , then T is (d) If D ✎ T ✑ is stratifiable with n ✛ t 1 ✛ t abelian. ☎ 1 ☎ 1 ☎ 1 the map τ : �✄✁✂✁✂✁✄� t n � t �✂✁✂✁✂✁✄� t 1 t 2 1 t n ✢ . 1 ✢✥✤✦ Perhaps 9 can be reduced to 8 in part (d). This is � n A diagonal group D ✎ T ✑ is primitive if and only if T is � 7 best possible since D ✎ Q 8 ✑ is generously transitive. characteristically simple. If T is simple, these are of � n We would like to have a similar bound for n if D ✎ T diagonal type in the O’Nan–Scott Theorem; otherwise they are of product type. is AS-friendly! 9 11 Diagonal groups with few simple factors � 1 D ✎ T ✑ : we nave Ω T , and the diagonal group is generated by right translations, automorphisms, and inversion. If we just use inner automorphisms and no inversion, we obtain a coherent configuration (the corresponding Schur ring is spanned by the conjugacy class sums); this is commutative, so fusing inverse pairs gives an association scheme. T 2 . The matrix with � 2 � u D ✎ T ✑ : We have Ω ✎ t ✑ entry ☎ 1 u is a Latin square. Any diagonal group preserves t the corresponding Latin square graph . So an AS-free diagonal group has at least four simple factors in its socle. 10