Math. J. Okayama Univ. 61 (2019), 199–204

TERWILLIGER ALGEBRAS OF SOME GROUPASSOCIATION SCHEMES
Nur Hamid and Manabu Oura
Abstract. The Terwilliger algebra plays an important role in the theory of association schemes. The present paper gives the explicit structures of the Terwilliger algebras of the group association schemes of thefinite groups P SL(2, 7), A6 , and S6 .

1. Introduction
Association schemes enable us to study combinatorial problems in a
unified way. We refer to [2, 6] for the foundations of association schemes.
In a series of papers [10, 11, 12], Terwilliger introduced a new method,
the so-called Terwilliger algebra, to investigate the commutative association
schemes. Since then there have been many investigations on Terwilligeralgebras (cf. [8, 7]). It is very important to know the explicit structure ofthe Terwilliger algebra. The cases of the group association schemes of S5and A5 were studied in [1] along the line of the work [3]. In the present
paper we determine the structures of the Terwilliger algebras of the groupassociation schemes of the finite groups P SL(2, 7), A6 , and S6 .
The computations were done with Magma [5] and SageMath [9].
2. Preliminaries
We begin with the definition of a group association scheme.
Definition 1. Let G be a finite group and C0 = {e}, C1 , . . . , Cd theconjugacy classes of G, where e is the identity of G. Define the relations
Ri (i = 0, 1, . . . , d) on G by(x, y) ∈ Ri ⇐⇒ yx−1 ∈ Ci .
Then X(G) = (G, {Ri }0≤i≤d ) forms a commutative association scheme ofclass d called the group association scheme of G.
We associate the matrix Ai of the relation Ri as{
1 if (x, y) ∈ Ri ,(Ai )x,y :=0 otherwise.
Mathematics Subject Classification. Primary 35C07; Secondary 35K57.
Key words and phrases. Terwilliger algebra, group association scheme.
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Then we have
Ai Aj =d∑pkij Ak

k=0and A0 , . . . , Ad generate the so-called Bose-Mesner algebra A.
Theintersection numbers pkij of the group association scheme X(G) are givenby|{(x, y) ∈ Ci × Cj |xy = z, z ∈ Ck }|.
The algebra A has a second basis E0 , . . . , Ed of primitive idempotents, and1 kEi ◦ Ej =q Ek ,|G| ij
where ◦ denotes Hadamard (entry-wise) multiplication. For each i =0, . . . , d, let Ei∗ and A∗i be the diagonal matrices of size |G| × |G| which aredefined as follows.
{
1, if x ∈ Ci
(Ei∗ )x,x :=(x ∈ G) ,0, if x ∈/ Ci(A∗i )x,x := |G|(Ei )e,x(x ∈ G).

Then E0∗ , . . . , Ed∗ form a basis for the dual Bose-Mesner algebra A∗ .

The
intersection numbers provide information for our structural results to follow.
We refer to the following relations [10].
Ei∗ Aj Ek∗ = 0 ⇔ pkij = 0 (0 ≤ i, j, k ≤ d),
k = 0 (0 ≤ i, j, k ≤ d).
Ei A∗j Ek = 0 ⇔ qij
We need to fix the ordering of the conjugacy classes. The following tablegives the representatives and the orders of conjugacy classes.
(1) P SL(2, 7)
C0
C1
C2rep.
(1)(357)(468)(2354786)|Ci | :
156
24C3C4C5rep. (2465837) (12)(34)(58)(67) (1235)(4876)|Ci | :
242142(2) A6C0C1C2C3C4rep. (1) (12)(34) (123) (123)(456) (1234)(56)|Ci |145404090
C5C6rep. (12345) (12346)
|Ci |
72
72

TERWILLIGER ALGEBRAS

201(3) S6C0C1C2C3(1)(12)(12)(34) (12)(34)(56)11545
15C4C5C6C7rep. (123) (123)(45) (123)(456)(1234)
40120
4090
C8C9C10(1234)(56) (12345) (123456)
90144
120
Finally we give the definition of the Terwilliger algebra of the groupassociation scheme. We shall denote by Mk the ring of k × k matrices
over the complex number C.
rep.
|Ci |

Definition 2. Let G be a finite group. The Terwilliger algebra T (G) of thegroup association scheme X(G) is a sub-algebra of M|G| generated by A andA∗ .
Since T (G) is closed under the conjugate-transpose, T (G) is semi-simple.
In the next section, we investigate the Terwilliger algebras of the groupassociation schemes of P SL(2, 7), A6 and S6 .
3. Results
In [1], Balmaceda and Oura gave the structures of the Terwilliger algebraof the group association schemes of S5 and A5 . Following their method, wedetermine the Terwilliger algebras for the cases P SL(2, 7), A6 , and S6 .
Theorem 3.1. The dimensions of T (P SL(2, 7)), A6 and T (S6) are given
as follows.
dim T (P SL(2, 7)) = 165,
dim T (A6 ) = 336,
dim T (S6 ) = 758.
Proof. We compute a set of linearly independent elements among Ei∗ Aj Ek∗∗ = E ∗ A E ∗ A E ∗ . By direct calculation we canand Ei∗ Aj Ek∗ · Ek∗ Al Em
i j k l m∗
see that any form Ei Ai1 Ej∗ · Ej∗ Ai2 Ek∗ · Ek∗ Ai3 El∗ linearly depends on theEi∗ Ai4 El∗ ’s and the Ei∗ Ai5 Ek∗1 · Ek∗1 Ai6 El∗ ’s. Therefore the products of morethan two elements of the form Ei∗ Aj Ek∗ give no new elements of a basis.
1□
1This answers a question raised by Prof.Terwilliger. Indeed our original argument hada gap. He informed us the reference [4]. Our result dim T (S6 ) = 758 is violated toConjecture 3.5.

202N. HAMID AND M. OURA

We provide the matrices below to show how many elements of a basis
occur. As these matrices are symmetric, we omit the entries below diagonal.
These matrices are indexed by the conjugacy classes in the order assumed
earlier. The entries of matrices indicate the dimension of each position.
For example, the entry 6 in the (C2 , C2 )-position for the group P SL(2, 7)
comes from the dimension of subspace that is the product of entry E2∗ Ai Ej∗
and Ek∗ Al E2∗ . The dimension coming from E2∗ Ai E2∗ is 5 and the product ofE2∗ Ai Ej∗ and Ek∗ Al E2∗ has dimension 6.

1 1 1 1 1 1

13 7 7 5 10 

6 6 3 6 
P SL(2, 7) : 
6 3 6 



4 5 
9


1 1 1 1 1 1 1

9 5 5 9 7 7 
8 8 9 8 8 
8 9 8 8 
A6 : 
16 13 13 

12 12 12

1 1 1 1 1 1 1 1 1 1 1

3 3 4 4 5 3 2 4 2 3 
646982646


8 8 8 7 4 8 4 8 
12 13 13 4 12 6 13 
19 16 3 13 6 12 
S6 : 

23313816
3 4 3 5 

12 6 13 



6 9 
19
We denote by Z(T (G)) the center of the Terwilliger algebra T (G) of afinite group G.
Lemma 3.2. The dimensions of Z(T (G)) for G = P SL(2, 7), A6 , S6 aregiven as follows.
dim Z(T (P SL(2, 7))) = 7,dim Z(T (A6 )) = 10,TERWILLIGER ALGEBRAS

203

dim Z(T (S6 )) = 14.
Proof. The result is obtained by determining a basis for the center. Wesolve a linear equation system∑ {xi y = yxi } ranging over all elements xi inthe basis of T (G) and y =cj bj , where bj are the basis elements of T (G)and cj is any scalar.
□∑ k
Let {ei : 1 ≤ i ≤ s} be a basis of Z(T (G)). Then we have ei ej =tij ekkand put Bi := (tij ) for 1 ≤ i ≤ s. Since these matrices mutually commute,they are simultaneously diagonalizable. We shall denote by v1 (i), . . . , vs (i)the diagonal entries of the diagonalized matrix of Bi and define the matrix
M by Mij := vi (j). Then we get the primitive central idempotents ε1 , . . . , εsby
(ε1 , . . . , εs ) = (e1 , . . . , es )M −1 .
Theorem 3.3. The degrees of the irreducible complex representations afforded by every idempotent are given below.
T (P SL(2, 7)) εi
ε1 ε2 ε3 ε4 ε5 ε6 ε7deg εi 1 2 3 3 5 6 9T (A6 )εiε1 ε2 ε3 ε4 ε5 ε6 ε7 ε8deg εi 1 3 3 4 4 6 6 7εiε9 ε10deg
8 10T (S6 )εiε1 ε 2 ε3 ε4 ε5 ε6 ε7 ε 8deg εi 1 1 1 3 3 4 6 7ε9 ε10 ε11 ε12 ε13 ε148 8 9 9 11 15
Proof. This is because that T (G)εi ∼= Mdi and that d2i = dim T (G)εi equals
the number of linearly independent elements in the set {xj εi }, where xj are
the basis elements of T .
□Theorems 3.1 and 3.3 are combined as
165 =12 + 22 + 32 + 32 + 52 + 62 + 92 ,336 =12 + 32 + 32 + 42 + 42 + 62 + 62 + 72 + 82 + 102 ,758 =12 + 12 + 12 + 32 + 32 + 42 + 62 + 72 + 82 + 82 + 92 + 92 + 112 + 152 .
The degrees of irreducible complex representations aﬀorded by every
primitive central idempotents enable us to get the following structure
theorem.
Corollary 3.4. We have that
T (P SL(2, 7)) ∼=M1 ⊕ M2 ⊕ M3 ⊕ M3 ⊕ M5 ⊕ M6 ⊕ M9 ,204

N. HAMID AND M. OURA

T (A6 ) ∼=M1 ⊕ M3 ⊕ M3 ⊕ M4 ⊕ M4 ⊕ M6 ⊕ M6 ⊕ M7 ⊕ M8⊕ M10 ,∼T (S6 ) =M1 ⊕ M1 ⊕ M1 ⊕ M3 ⊕ M3 ⊕ M4 ⊕ M6 ⊕ M7 ⊕ M8⊕ M8 ⊕ M9 ⊕ M9 ⊕ M11 ⊕ M15 .
Acknowledgment. The second named author was supported by JSPS
KAKENHI Grant Number JP25400014. The authors would like to thankProf. Terwilliger for helpful discussions.
References[1] J. M. P. Balmaceda, M. Oura, The Terwilliger algebras of the group association
schemes of S5 and A5 , Kyushu J. Math., 48, no. 3 (1994), 221–231.
[2] E. Bannai, T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, California, 1984.
[3] E. Bannai, A. Munemasa, The Terwilliger algebras of group association schemes,Kyushu J. Math., 49, no. 1 (1995), 93–102.
[4] M. M. Bogaerts, P. Dukes, Semidefinitie programming for permutation codes, Discrete
Mathematics, 326, (2014), 34–43.
[5] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language,
J. Symbolic Comput., 24, no. 3-4 (1997), 235–265.
[6] A. E. Brouwer, C. M. Cohen, A. Neumaier, Distance-regular graphs, Ergebnisse derMathematik und ihrer Grenzgebiete (3), 18. Springer-Verlag, Berlin, 1987.
[7] T. Ito, Tridiagonal pairs and q-Onsager algebras(Japanese), SIAM J. Math. Anal.,Sūgaku 65, no. 1 (2013), 69–92.
[8] W. Martin, H. Tanaka, Commutative association schemes, European J. Combin., 30,no. 6 (2009), 1497–1525.
[9] The Sage Developers, Sagemath, the Sage Mathematics Software System (Version6.4.1), http://www.sagemath.org, 2014.
[10] P. Terwilliger, The subconstituent algebra of an association scheme I, J. Algebraic
Combin., 1, no. 4 (1992), 363–388.
[11] P. Terwilliger, The subconstituent algebra of an association scheme II, J. Algebraic
Combin., 2, no. 1 (1993), 73–103.
[12] P. Terwilliger, The subconstituent algebra of an association scheme III, J. Algebraic
Combin., 2, no. 2 (1992), 177–210.
Nur Hamid
Faculty of Mathematics and Physics, Kanazawa University
andNurul Jadid University, Probolinggo, Indonesia
e-mail address: hamidelfath@gmail.com
Manabu Oura
Faculty of Mathematics and Physics, Kanazawa University
e-mail address: oura@se.kanazawa-u.ac.jp(Received October 27, 2017 )(Accepted November 8, 2018 )