Group theoretic aspects of the theory of association schemes

Transcription

1 Group theoretic aspects of the theory of association schemes Akihiro Munemasa Graduate School of Information Sciences Tohoku University October 29, 2016 International Workshop on Algebraic Combinatorics Anhui University A. Munemasa (Tohoku University) association schemes Hefei / 18

2 Association schemes Bannai and Ito, Algebraic Combinatorics I (1984). P- and Q-polynomial schemes A. Munemasa (Tohoku University) association schemes Hefei / 18

3 Association schemes Bannai and Ito, Algebraic Combinatorics I (1984). P- and Q-polynomial schemes designs and codes (Delsarte theory) A. Munemasa (Tohoku University) association schemes Hefei / 18

4 Association schemes Bannai and Ito, Algebraic Combinatorics I (1984). P- and Q-polynomial schemes designs and codes (Delsarte theory) spherical designs A. Munemasa (Tohoku University) association schemes Hefei / 18

5 Association schemes Bannai and Ito, Algebraic Combinatorics I (1984). P- and Q-polynomial schemes Terwilliger algebras designs and codes (Delsarte theory) spherical designs A. Munemasa (Tohoku University) association schemes Hefei / 18

6 Association schemes Bannai and Ito, Algebraic Combinatorics I (1984). P- and Q-polynomial schemes Terwilliger algebras designs and codes (Delsarte theory) SDP spherical designs A. Munemasa (Tohoku University) association schemes Hefei / 18

7 Association schemes Bannai and Ito, Algebraic Combinatorics I (1984). P- and Q-polynomial schemes Terwilliger algebras designs and codes (Delsarte theory) SDP spherical designs Euclidean designs A. Munemasa (Tohoku University) association schemes Hefei / 18

8 Permutation representations association scheme multiplicity-free permutation representation ordinary representation theory A. Munemasa (Tohoku University) association schemes Hefei / 18

9 Permutation representations association scheme multiplicity-free permutation representation ordinary representation theory inequivalent permutation representations which are equivalent as representations A. Munemasa (Tohoku University) association schemes Hefei / 18

10 Permutation representations association scheme multiplicity-free permutation representation ordinary representation theory inequivalent permutation representations which are equivalent as representations (like cospectral graphs) A. Munemasa (Tohoku University) association schemes Hefei / 18

11 Permutation representations association scheme multiplicity-free permutation representation ordinary representation theory inequivalent permutation representations which are equivalent as representations (like cospectral graphs) ordinary representation centralizer = algebra permutation representation combinatorial object A. Munemasa (Tohoku University) association schemes Hefei / 18

12 Centralizer Multiplicity-free permutation representation of a finite group G acting on a set X centralizer algebra has a basis consisting of (0, 1)-matrices A 0 = I, A 1,..., A d with d A i = J. i=0 These matrices represent orbitals: X X = d i=0 R i (G-orbits). A. Munemasa (Tohoku University) association schemes Hefei / 18

13 Association scheme Since it is a basis of an algebra, A i A j = d pija k k. (1) k=0 {A 0 = I, A 1,..., A d } = {A 0 = I, A 1,..., A d }. (2) This allows one to forget the group G to define an association scheme: (X, {R i } d i=0 ); where we require the matrices representing the partition {R i } d i=0 of X X satisfy (1) and (2). A. Munemasa (Tohoku University) association schemes Hefei / 18

14 Association scheme Since it is a basis of an algebra, A i A j = d pija k k. (1) k=0 {A 0 = I, A 1,..., A d } = {A 0 = I, A 1,..., A d }. (2) This allows one to forget the group G to define an association scheme: (X, {R i } d i=0 ); where we require the matrices representing the partition {R i } d i=0 of X X satisfy (1) and (2). multiplicity-free = commutative. A. Munemasa (Tohoku University) association schemes Hefei / 18

15 Krein parameters q k ij 0 Multiplicity-free permutation representation of a finite group G acting on a set X has centralizer algebra A = A 0 = I, A 1,..., A d V = C X = L(X ) decomposes as C[G]-module: V = V 0 V 1 V d V 0 =constant. The orthogonal projections E i : V V i form another basis of A, so E i E j = 1 X d qij k E k. i=0 A. Munemasa (Tohoku University) association schemes Hefei / 18

16 Scott s theorem V = V 0 V 1 V d, θ = χ 0 + χ χ d permutation character E i : V V i : the orthogonal projection, E i E j = 1 X Theorem (Scott (1977)) q k ij 0 = (χ i χ j, χ k ) 0. d qij k E k. i=0 A. Munemasa (Tohoku University) association schemes Hefei / 18

17 q k ij 0 = (χ i χ j, χ k ) 0 Problem (Bannai and Ito, p.130) To what extent is the converse of Scott s theorem true? A counterexample: Ind S 2n S n S n (Johnson scheme J(2n, n)). A. Munemasa (Tohoku University) association schemes Hefei / 18

18 q k ij 0 = (χ i χ j, χ k ) 0 Problem (Bannai and Ito, p.130) To what extent is the converse of Scott s theorem true? A counterexample: Ind S 2n S n S n (Johnson scheme J(2n, n)). But this is imprimitive: S n S n S n S 2 S 2n. A. Munemasa (Tohoku University) association schemes Hefei / 18

19 q k ij 0 = (χ i χ j, χ k ) 0 Problem (Bannai and Ito, p.130) To what extent is the converse of Scott s theorem true? A counterexample: Ind S 2n S n S n (Johnson scheme J(2n, n)). But this is imprimitive: S n S n S n S 2 S 2n. Is there a primitive counterexample? A. Munemasa (Tohoku University) association schemes Hefei / 18

20 q k ij = 0 but (χ i χ j, χ k ) 0? The primitive counterexample of the smallest degree is of degree 66. G = PGL(2, 11) acting on PGL(2, 11)/D 20 A. Munemasa (Tohoku University) association schemes Hefei / 18

21 Eigenvalues of association schemes Multiplicity-free permutation representation of a finite group G acting on a set X = G/H has centralizer algebra A = A 0 = I, A 1,..., A d V = C X = L(X ) decomposes as C[G]-module: V = V 0 V 1 V d V 0 =constant. A i acts on V j as a scalar: 1 H g Ha i H χ j (g). A. Munemasa (Tohoku University) association schemes Hefei / 18

22 Eigenvalues of association schemes Multiplicity-free permutation representation of a finite group G acting on a set X = G/H has centralizer algebra A = A 0 = I, A 1,..., A d Eigenvalues of A i are { 1 H g Ha i H χ j (g) 0 j d} Q(exp 2π 1 e(g) ). where e(g) is the exponent of G. A. Munemasa (Tohoku University) association schemes Hefei / 18

23 Splitting fields of association schemes Question (Bannai and Ito, p.123) Are there any association schemes in which eigenvalues of A i s are not all in a cyclotomic number field? By the Kronecker Weber theorem, this is equivalent to ask whether the Galois group of the splitting field of the characteristic polynomial of A i is abelian. Theorem (M. (1991), Coste Gannon (1994)) If q k ij Q, then eigenvalues of A i s are in a cyclotomic number field. A. Munemasa (Tohoku University) association schemes Hefei / 18

24 Unlike the converse to Scott s theorem, the question: Question (Bannai and Ito, p.123) Are there any association schemes in which eigenvalues of A i s are not all in a cyclotomic number field? makes sense for non-commutative association schemes as well. A. Munemasa (Tohoku University) association schemes Hefei / 18

25 Splitting fields of association schemes Let A = A 0 = I, A 1,..., A d be an algebra spanned by disjoint (0, 1)-matrices. 1 A is the centralizer of a multiplicity-free permutation representation, then the formula for spherical functions implies that eigenvalues of A i s are cyclotomic. 2 Bannai Ito asks: the same holds if A is commutative (without group action)? 3 A is the centralizer of a non-multiplicity-free permutation representation, then this is not true. A. Munemasa (Tohoku University) association schemes Hefei / 18

26 Non-multiplicity-free permutation group A. Ryba and S. Smith: of degree 165. G = PGL(2, 11) acting on PGL(2, 11)/D 8 A. Munemasa (Tohoku University) association schemes Hefei / 18

27 Non-multiplicity-free permutation group A. Ryba and S. Smith: G = PGL(2, 11) acting on PGL(2, 11)/D 8 of degree 165 imprimitive. qij k = 0 but (χ i χ j, χ k ) 0? The primitive example of the smallest degree is G = PGL(2, 11) acting on PGL(2, 11)/D 20 of degree 66. A. Munemasa (Tohoku University) association schemes Hefei / 18

28 Non-cyclotomic numbers 3 = e π 1/6 + e π 1/6 A. Munemasa (Tohoku University) association schemes Hefei / 18

29 Non-cyclotomic numbers 3 = e π 1/6 + e π 1/6 4 3 / cyclotomic field A. Munemasa (Tohoku University) association schemes Hefei / 18

30 Non-cyclotomic numbers 3 = e π 1/6 + e π 1/6 If 4 3 / cyclotomic field has an irrational eigenvalue, then A is likely to have a non-cyclotomic eigenvalue. A 2 A. Munemasa (Tohoku University) association schemes Hefei / 18

31 Non-cyclotomic numbers 3 = e π 1/6 + e π 1/6 If A 2 ci 4 3 / cyclotomic field (distance-2 graph) has an irrational eigenvalue, then A is likely to have a non-cyclotomic eigenvalue. A. Munemasa (Tohoku University) association schemes Hefei / 18

32 Non-multiplicity-free permutation group A. Ryba and S. Smith: G = PGL(2, 11) acting on PGL(2, 11)/D 24 has as an eigenvalue, which is a bipartite half of a bipartite 3-regular graph (flag-transitive incidence structure) on vertices. A. Munemasa (Tohoku University) association schemes Hefei / 18

33 Non-multiplicity-free permutation group A. Ryba and S. Smith: G = PGL(2, 11) acting on PGL(2, 11)/D 24 has as an eigenvalue, which is a bipartite half of a bipartite 3-regular graph (flag-transitive incidence structure) on vertices. = is an eigenvalue of A 2 A. Munemasa (Tohoku University) association schemes Hefei / 18

34 Non-multiplicity-free permutation group A. Ryba and S. Smith: G = PGL(2, 11) acting on PGL(2, 11)/D 24 has as an eigenvalue, which is a bipartite half of a bipartite 3-regular graph (flag-transitive incidence structure) on vertices. = is an eigenvalue of A 2 = is an eigenvalue of A A. Munemasa (Tohoku University) association schemes Hefei / 18

35 Non-multiplicity-free permutation group A. Ryba and S. Smith: G = PGL(2, 11) acting on PGL(2, 11)/D 24 has as an eigenvalue, which is a bipartite half of a bipartite 3-regular graph (flag-transitive incidence structure) on vertices. = is an eigenvalue of A 2 = is an eigenvalue of A = is an eigenvalue of the line graph (G acts transitively, imprimitively). A. Munemasa (Tohoku University) association schemes Hefei / 18

36 Primitive counterexample 1 The smallest primitive group with non-cyclotomic eigenvalue is PSL(2, 19)/D The smallest (imprimitive) transitive group with non-cyclotomic eigenvalue is of degree 32 (due to classification of association schemes by Hanaki). 3 Hanaki and Uno (2006). Even for a prime number of points, the question is unsettled. Thank you for your attention! A. Munemasa (Tohoku University) association schemes Hefei / 18