Nonexistence of Exceptional 5-class Association Schemes with Two Q -polynomial Structures
aa r X i v : . [ m a t h . C O ] N ov Nonexistence of Exceptional 5-class Association Schemes with Two Q -polynomialStructures Jianmin Ma a , Kaishun Wang b a Hebei Key Lab of Computational Mathematics & ApplicationsandCollege of Math & Info. SciencesHebei Normal University, Shijiazhuang 050016, China b Sch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, Beijing 100875, China
Abstract
In [H. Suzuki, Association schemes with multiple Q -polynomial structures, J. Algebraic Combin. 7 (1998) 181-196],Suzuki gave a classification of association schemes with multiple Q -polynomial structures, allowing for one exceptional casewhich has five classes. In this paper, we rule out the existence of this case. Hence Suzuki’s theorem mirrors exactly the well-known counterpart for association schemes with multiple P -polynomial structures, a result due to Eiichi Bannai and EtsukoBannai in 1980. Keywords:
Association scheme, P - or Q -polynomial structure, Fusion scheme
1. Introduction
Eiichi Bannai and Etsuko Bannai [2] studied association schemes with multiple P -polynomial structures and obtained thefollowing classification. See [2] or [1, Section III.4]. Theorem 1.
Suppose that X is a symmetric association scheme with a P-polynomial structure A , A , . . . , A d for the adjacencymatrices. If X is not a polygon and has another P-polynomial structure, then the new structure is one of the following: (I) A , A , A , A , . . . , A , A , A ;(II) A , A d , A , A d − , A , A d − , A , A d − , . . . ;(III) A , A d , A , A d − , A , A d − , . . . , A d − , A , A d − , A , A d − , A ;(IV) A , A d − , A , A d − , A , A d − , . . . , A , A d − , A , A d − , A , A d .Hence, X admits at most two P-polynomial structures. The parametric conditions for each case above can be found in [1, Section III.4]. The question was raised whether similarresult could be obtained for association schemes with multiple Q-polynomial structures in [1, Section III.4] and [2]. Suzuki [7]settled this question in the following theorem in 1998.
Theorem 2.
Suppose that X is a symmetric association scheme with a Q-polynomial structure E , E , . . . , E d for the primitiveidempotents. If X is not a polygon and has another Q-polynomial structure, then the new structure is one of the following: (I) E , E , E , E , . . . , E , E , E ;(II) E , E d , E , E d − , E , E d − , E , E d − , . . . ;(III) E , E d , E , E d − , E , E d − , . . . , E d − , E , E d − , E , E d − , E ;(IV) E , E d − , E , E d − , E , E d − , . . . , E , E d − , E , E d − , E , E d ;(V) d = and E , E , E , E , E , E .Hence, X admits at most two Q-polynomial structures. Note that case (V) has no counterpart in Theorem 1. In fact, its counterpart A , A , A , A , A , A did appear in the originalstatement of Theorem 1 in [2] but was eliminated with an easy combinatorial argument. It has been wondered if this case canalso be eliminated, e.g. [5, p.1506]. We will do exactly this in this paper, and so Theorems 1 and 2 are dual to each other. Theorem 3 (Main theorem).
Case (V) in Theorem 2 does not occur.
Email addresses:
[email protected] (Jianmin Ma), [email protected] (Kaishun Wang)
1e conclude this section with the outline of our proof. Any automorphism of the splitting field for an association schemeinduces a permutation of its primitive idempotents. In particular, when applied to a given Q -polynomial structure, each nontriv-ial automorphism induces another Q -polynomial structure. By Theorem 2, the Galois group has order at most 2. If the Galoisgroup has order 2 and fixes the Krein parameters, we can compare the Krein parameters from the two Q -polynomial structuresand then determine the Krein matrix B ∗ for a putative association scheme X in case (V). It turns out that B ∗ is completelydetermined by the first multiplicity m . X has a fusion scheme with 3 classes, whose parameters can be obtained from B ∗ . Usingelementary number theory we determine the possible values for m and further show that these values give infeasible parametersof the original scheme. If the Galois group is trivial or it has order 2 but does not fix the Krein parameters, we will derive twoequations from the two Q -polynomial structures, which lead to a contradiction. All calculations are verified with the softwarepackage Maple 14 .All association schemes in this paper are symmetric. The reader is referred to [1, 8] for missing definitions. For recentactivities on P - or Q -polynomial association schemes and related topics, see the recent survey paper by Martin and Tanaka [5].
2. Preliminaries
In this paper, we adopt the notation in [1]. Let X be a symmetric association scheme with adjacency matrices A i andprimitive idempotents E i , 0 ≤ i ≤ d . Then A , . . . A d span an algebra M over the real field R , called the Bose-Mesner algebraof X .The intersection numbers p ki j and Krein parameters q ki j are defined by A i A j = d X k = p ki j A k and E i ◦ E j = | X | − d X k = q ki j E k , where ◦ is the entry-wise (Hadamard) product. The intersection matrix B i has ( j , k )-entry p ki j and the Krein matrix B ∗ i has( j , k )-entry q ki j . When there is a possibility of confusion, we write p ki , j for p ki j and do the same for Krein parameters.The adjacency matrices A i and primitive idempotents E i form two bases for M and so we can write down the transitionmatrices between them: A i = d X j = p i ( j ) E j and E i = | X | − d X j = q i ( j ) A j . The numbers p i ( j ) and q i ( j ) are called eigenvalues and dual eigenvalues of X . Set k i = p i (0) and m i = q i (0); k i is calledthe valency of A i and m i is the multiplicity of E i . Set P = [ p i ( j )], and Q = [ q i ( j )], whose ( j , i )-entries are p i ( j ) and q i ( j ),respectively. P and Q are called the first eigenmatrix and the second eigenmatrix of X .We call A , A , . . . , A d a P-polynomial structure for X if the following conditions are satisfied: for 0 ≤ i , j , k ≤ d ,(P1) p ki j = i , j , k is greater than the sum of other two;(P2) p ki j , i , j , k is equal to the sum of other two.In this case we write c i = p i , i − , a i = p i i , and b i = p i , i + .Dually, we call E , E , . . . , E d a Q-polynomial structure for X if, for 0 ≤ i , j , k ≤ d ,(Q1) q ki j = i , j , k is greater than the sum of other two;(Q2) q ki j , i , j , k is equal to the sum of other two.Similarly, we write c ∗ i = q i , i − , a ∗ i = q i i , b ∗ i = q i , i + . Note b ∗ = m , c ∗ =
1, and by (Q2), c ∗ i , ≤ i ≤ d ) and b ∗ i , ≤ i ≤ d −
1) .An association scheme X is called P - (resp. Q -) polynomial if it has at least one P - (resp. Q -) polynomial structure. Fora P -polynomial scheme, each B i is a polynomial of B of degree i . Similar assertion holds for Q -polynomial schemes; moreprecisely, B ∗ i = ( B ∗ B ∗ i − − a ∗ i − B ∗ i − − b ∗ i − B ∗ i − ) / c ∗ i , ≤ i ≤ d , (1)where B ∗ = I , the identity matrix of order d + X is the Galois extension L of the rational field Q by adjunction of all theeigenvalues (hence dual eigenvalues) of X ([1, Section II.7], [6]). Each automorphism of L / Q applies entry-wise to all matricesin the Bose-Mesner algebra. Let K be the extension of Q by adjunction of all the Krein parameters. Since Krein parametersare rational functions of the eigenvalues [1, p.65], K is an intermediate field between Q and L . If the Galois group Gal( L / Q ),denoted also by Gal( X ), is non-trivial, then X has irrational eigenvalues. For any σ ∈ Gal( X ), E σ , E σ , . . . , E σ d are all primitiveidempotents of M and thus a permutation of E , E , . . . , E d . It is not hard to see that Gal( X ) acts faithfully on the primitiveidempotents. In particular, if E , E , . . . , E d is a Q -polynomial structure, then E σ , E σ , . . . , E σ d is also a Q -polynomial structure.One can build new association schemes from existing ones by merging classes. For a scheme X = ( X , { R i } di = ), anotherscheme Y = ( X , { S j } ej = ) is called a fusion of X if each S j is a union of some subset of relations R i . For any subgroup2 ≤ Gal( X ), the orbits of H on the primitive idempotents of X give rise to a fusion scheme. The parameters of Y can beobtained from these of X [3].Now we close this section with parametric conditions for case (V) in Theorem 2, which will be needed later. Lemma 4. [7]
Theorem 2 (V) holds if and only if q = q = q = q = , q and q = .
3. Proof of the main theorem
Let X be an association scheme with two Q -polynomial structures in Theorem 2 (V). In the rest of this paper, E , E , . . . , E is a fixed Q -polynomial structure for X . Let B ∗ be the first Krein matrix of X . Now by Lemma 4, we may assume B ∗ = m a ∗ c ∗ b ∗ a ∗ c ∗ b ∗ a ∗ c ∗ b ∗ a ∗ mb ∗ : = ∗ c ∗ c ∗ c ∗ m a ∗ a ∗ a ∗ a ∗ m b ∗ b ∗ b ∗ b ∗ ∗ , where we use the notation of [1, p.189] for tridiagonal matrices. Note the columns of B ∗ sum to the rank m ( = m ) of E .We use a hat ˆ to indicate the second Q -polynomial structure. Theorem 2 says E ˆ0 = E , E ˆ1 = E , E ˆ2 = E , E ˆ3 = E , E ˆ4 = E , E ˆ5 = E . (2)By the remarks from the previous section, | Gal( X ) | ≤ Q ⊆ K ⊆ L . So we have either Q = K , L or K = L . Now weprove the main theorem under two separate assumptions: A) K , L ; B) K = L . , L In this subsection, we assume K , L and hence Q = K . In this case, Gal( X ) is generated by an involution σ , i.e., σ is theidentity. Now E σ , E σ , E σ , E σ , E σ , E σ is the other Q-polynomial structure and thus E σ k = E ˆ k . For brevity we write ˆ q rst : = q ˆ r ˆ s ˆ t . Since K = Q , σ fixes the Krein parameters, i.e., ˆ q rst = q rst . (3)Since m i = q ii , we have m = m ˆ2 ( = m ) by (3). It follows that b ∗ = c ∗ , since m i b ∗ i = m i + c ∗ i + . Since q = ˆ q , a ∗ = b ∗ = m −
1. Similarly, a ∗ = q = ˆ q =
0. Since B ∗ i is a polynomial in B ∗ of degree i , we use Eq. (1) to calculate B ∗ : B ∗ = m − a ∗ c ∗ m − mc ∗ ( m − a ∗ c ∗ c ∗ ( m − + a ∗ + c ∗ − mc ∗ a ∗ c ∗ c ∗ c ∗ c ∗ c ∗
00 ( m − c ∗ c ∗ a ∗ c ∗ c ∗ c ∗ + b ∗ c ∗ − mc ∗ a ∗ c ∗ c ∗ c ∗ mc ∗ b ∗ b ∗ c ∗ a ∗ b ∗ c ∗ c ∗ b ∗ + a ∗ + mb ∗ − mc ∗ ma ∗ c ∗ b ∗ b ∗ c ∗ a ∗ b ∗ c ∗ m ( b ∗ − c ∗ . By Lemma 4, q = b ∗ = B ∗ = B ∗ ˆ3 . Now compute B ∗ ˆ3 with (1) and compare B ∗ ˆ3 with B ∗ to obtain the following: b ∗ = c ∗ c ∗ , a ∗ = a ∗ = c ∗ c ∗ , mc ∗ = c ∗ ( m − . Using the fact that the columns of B ∗ sum to m , we can obtain B ∗ = ∗ m −
12 2 mm + m − m + m m − m +
1) 0 ( m − m + m m − mm + m ( m − m + ∗ . | Gal( X ) | = { E } , { E , E } , { E , E } , { E } are the orbits of Gal( X ) on the primitive idempotents. By the remark beforeLemma 4, these orbits give rise to a fusion scheme Y . Let T = { } , T = { , } , T = { , } , T = { } . Then the Krein parameters s ki j of Y can be calculated from q rst : s γ i j = X α ∈ T i , β ∈ T j q γαβ (4)is independent of the choice γ in T k .Using (4), we can obtain the Krein matrix C ∗ of Y : C ∗ = m m −
12 20 m − m + m + m + m − m + m ( m − mm + m − m + . Now we can obtain the second eigenmatrix S of Y from C ∗ : S = m m m − m − m + + ∆ m +
1) (5 m − m + + ∆ )( m − m + − (3 m − m + + ∆ ) m m + m + − m − ∆ m +
1) (5 m − m + − ∆ )( m − m + − (3 m − m + − ∆ ) m m + , where ∆ = p ( m − m + m − m + . The first eigenmatrix of Y is ( m + m + S − , and its top-most row are thevalencies of Y : 1 , m and m (7 m − m + p ( m − m + m − m + ± m . (5)In the rest of this subsection, we determine the possible values of m . Since a valency is a positive integer, the ratio m (7 m − m + p ( m − m + m − m +
1) (6)is an integer, and in addition the following must hold: (i) p ( m − m + m − m +
1) divides m (7 m − m + m − m + m − m +
1) is a perfect square.If the ratio (6) is an integer, then any prime p which divides m − m + m or 7 m − m +
7. If p divides m then clearly p =
3. If p divides 7 m − m + p divides7 m − m + − (7 m − m + = m + . Hence either p = p divides m + . In the second case, p divides( m − m + − ( m + = ( m − m − . Now p divides 8 if m ≡ p divides 9 if m ≡ m − + a b . The greatest commondivisor of 7 m − m + m − m + p is an odd prime which divides both 7 m − m + m − m +
1, then (subtracting the first from the second and dividing by 2), p divides m + m −
3. Then (subtracting the firstequation from 7 times the new equation), p divides 92 m − = m −
7) and hence p divides 7 m + m = m − m + + (23 m − . Since p does not divide m , p divides 7 m +
1, and thus divides (23 m − − m + = m − . But now p also divides 7( m − + p divides 36 and p = m − m + a b and 9 m − m + e f for integers a , b , e , f . But nownotice that the greatest common divisor d of m − m + m − m + m −
80, which is the latter equationminus 9 times the former. Also, 16 does not divide d , since m − m + = ( m − + m − m + = m + ( m − ). If h is the odd part of d , then m ≡ h divides 24. Hence d divides 24, since m − m + = ( m − + m − +
24. If m is divisible by 3, then 9 m − m + m = m is not divisible by 3. If 9 divides m − m + m ≡ m − m + ∈ { , , } , a contradiction. Hence m − m + =
24 and m = m − m + = m = m not divisible by 3, as we are currently assuming). Thus m = m = c ∗ = ( m − / > m =
5. Hence B ∗ is determined. Now it is a routine exercise to obtain the second eigenmatrix Q from B ∗ : Q = − − − + √
213 2(1 − √ + √ − − √ − − √
213 2(1 + √ − √ − + √ + √
213 2(4 + √ − √ −
253 1 − √ − √
213 2(4 − √ + √ −
253 1 + √ . We can obtain the intersection numbers from the matrix Q by [1, Th.3.6 (ii), p.65]: B = . Since intersection numbers must be integers, we reach a contradiction. We complete the proof of our main theorem under theassumption K , L . = L In this subsection, we assume K = L . So Gal( X ) is either trivial or Gal( X ) has order 2 but its nonidentity element does notfixed all Krein parameters. So relation (3) no longer holds.The (5 ,
5) entry in B ∗ ˆ5 is q ˆ5ˆ5ˆ5 = a ∗ and by Lemma 4, a ∗ = b ∗ = m −
1. So we begin with B ∗ = ∗ c ∗ c ∗ c ∗ m a ∗ a ∗ a ∗ m m − b ∗ b ∗ b ∗ ∗ . Now compute B ∗ with (1). We obtain q = m ( b ∗ − c ∗ and so b ∗ = B ∗ , B ∗ , B ∗ , and B ∗ . Withthe second Q -polynomial structure, the first Krein matrix B ∗ ˆ1 = ( q ˆ k ˆ1ˆ i ) with ( i , k ) entry q ˆ k ˆ1ˆ i is tridiagonal with nonzero o ff -diagonalentries. The (2,1) entry of B ∗ ˆ1 is ma ∗ c ∗ c ∗ , so a ∗ ,
0. We also have q ˆ4ˆ1ˆ1 = c ∗ b ∗ + a ∗ − a ∗ a ∗ − ( m − c ∗ − a ∗ a ∗ c ∗ c ∗ c ∗ and q ˆ4ˆ1ˆ2 = c ∗ b ∗ + a ∗ − a ∗ ∗ a ∗ − ( m − c ∗ c ∗ c ∗ , a ∗ =
0. Now, we have B ∗ = ∗ c ∗ c ∗ c ∗ m a ∗ a ∗ m m − b ∗ b ∗ ∗ . Similarly, we compute B ∗ , from which we can obtain B ∗ ˆ1 . We findˆ q = a ∗ m − a ∗ c ∗ b ∗ − b ∗ a ∗ c ∗ c ∗ c ∗ c ∗ . By Lemma 4, q =
0. As ˆ q = p , we have a ∗ m − a ∗ c ∗ b ∗ − b ∗ a ∗ c ∗ = . (7)Now we derive another equation. Let v ∗ i ( x ) (2 ≤ i ≤
5) be the polynomials defined by B ∗ [1, Section III.1]: v ∗ ( x ) = , v ∗ ( x ) = x , xv ∗ i ( x ) = b ∗ i − v ∗ i − ( x ) + a ∗ i v ∗ i ( x ) + c ∗ i + v ∗ i + ( x ) , where c ∗ : =
1. So v ∗ ( x ) annihilates B ∗ . Since m is an eigenvalue of B ∗ , we have v ∗ ( m ) = m ( m − − m a ∗ + ma ∗ c ∗ − mb ∗ c ∗ + ma ∗ a ∗ + a ∗ b ∗ c ∗ + a ∗ b ∗ c ∗ + c ∗ b ∗ c ∗ ) c ∗ c ∗ c ∗ = , which gives − m a ∗ + ma ∗ c ∗ − mb ∗ c ∗ + ma ∗ a ∗ + a ∗ b ∗ c ∗ + a ∗ b ∗ c ∗ + c ∗ b ∗ c ∗ = . Using Eq. (7), it simplifies to − m a ∗ + ma ∗ c ∗ − mb ∗ c ∗ + ma ∗ a ∗ + ma ∗ + c ∗ b ∗ c ∗ = . By a ∗ + c ∗ = m − b ∗ , it becomes ma ∗ (1 − b ∗ ) = b ∗ c ∗ ( m − c ∗ ) . (8)Taking the di ff erence of Eq. (7) and (8), we obtain ma ∗ b ∗ = b ∗ c ∗ ( a ∗ + c ∗ − m ) + a ∗ b ∗ c ∗ . This simplifies to a ∗ b ∗ b ∗ = − b ∗ b ∗ c ∗ , as c ∗ + b ∗ = m . Hence, a ∗ + c ∗ =
0, absurd. We have proved the main theorem under theassumption K = L .
4. Concluding RemarksRemark 1. P -polynomial structures (I) and (II) come in pair. As pointed out in [1, p.243], (I) and (II) are dual to eachother, i.e., exchanging the roles of the first and second structures, the first is of type (II) (resp. (I)) in terms of the second ifthe second is of type (I) (resp. (II)) in terms of the first. The structures (III), (IV) are self dual. The same comment applies to Q -polynomial structures. Remark 2.
Cases (I), (II) and (IV) of Theorem 1 are realized [1, Section III.4]. Similarly for Theorem 2, case (I) isrealized by the half cube H (2 d + , e H (2 d + ,
2) and the dual polar graphon [ A d − ( r )] , r ≥
2, and case (IV) by the cube H ( d , d even. Note each member of these four families is a distance regulargraph admitting two Q -polynomial structures. Dickie [4] classified such graphs with diameter d ≥ k ≥
3. Inview of Remark 1, H (2 d + ,
2) admits a Q -polynomial structure of type (II); e H (2 d + ,
2) and [ A d − ( r )] , r ≥ , admit a Q -polynomial structure of type (I).Suzuki [9] showed that the class number d ≤ d ( ≥
3) classes, which admits two Q -polynomial structures and no P -polynomial structures. Remark 3.
A careful reader will notice that the proof in Subsection 3.2 does not even use the assumption K = L , soSubsection 3.1 can be omitted. However, we feel that Subsection 3.1 serves to illustrate a powerful use of the Galois group.Plus, the elementary number theoretic proof in this subsection is very beautiful.6 cknowledgments We are indebted to the anonymous reviewers whose detailed and thoughtful reports save us a few mistakes and improvethe readability of this paper. We thank Hajime Tanaka for bringing to our attention the problem of this paper during the 2010Workshop on Schemes and Spheres at Worcester Polytechnic Institute, and thus the first author thanks Bill Martin for hishospitality and many inspiring conversations. Geo ff Robinson provides the elementary approach in Section 3 which determinesthe values of m from (6). We also thank Eiichi Bannai for sending us his original paper [2] that helps to correct a few inaccuratestatements in an early draft.JM is partially supported by the Natural Science Foundation of Hebei province (A2012205079) and Science Foundation ofHebei Normal University (L2011B02). KW is partially supported by NSF of China (11271047, 11371204) and the FundamentalResearch Funds for the Central Universities of China. References [1] E. Bannai, T. Ito, Algebraic Combinatorics I:Association Schemes, The Benjamin / Cummings Publishing Company, Inc., Menlo Park, CA, 1984.[2] E. Bannai, E. Bannai, How many P -polynomial structures can an association scheme have?, European J. Combin. 1 (1980) 289–298.[3] E. Bannai, S. Y. Song, Character tables of fission schemes and fusion schemes, European J. Combin. 14 (1993) 385–396.[4] G. A. Dickie, Q-polynomial structures for association schemes and distance regular graphs, PhD thesis, University of Wisconsin-Madison, 1995.[5] W. J. Martin, H. Tanaka, Commutative association schemes, European J. Combin. 30 (2009) 1497–1525.[6] A. Munemasa. Splitting fields of association schemes, J. Combin. Theory Ser. A 57 (1991), 157–161.[7] H. Suzuki, Association schemes with multiple Q -polynomial structures, J. Algebraic Combin. 7 (1998) 181–196.[8] H. Suzuki, Imprimitive Q -polynomial association schemes, J. Algebraic Combin. 7(1998) 165–180.[9] H. Suzuki, A Note on Association Schemes with Two P-Polynomial Structures of Type III, J. Combin. Theory, Ser. A 74(1996) 158–168.-polynomial association schemes, J. Algebraic Combin. 7(1998) 165–180.[9] H. Suzuki, A Note on Association Schemes with Two P-Polynomial Structures of Type III, J. Combin. Theory, Ser. A 74(1996) 158–168.