¿^creative ^commor
ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 51-60
https://doi.org/10.26493/1855-3974.2034.dad
(Also available at http://amc-journal.eu)
Association schemes with a certain type of p-subschemes*
Wasim Abbas f ©, Mitsugu Hirasaka
Department of Mathematics, College of Sciences, Pusan National University, 63 Beon-gil 2, Busandaehag-ro, Geumjung-gu, Busan 609-735, Korea
Received 4 July 2019, accepted 16 July 2020, published online 2 November 2020 Abstract
In this article, we focus on association schemes with some properties derived from the orbitals of a transitive permutation group G with a one-point stabilizer H satisfying
H < Ng(H ) < Ng(Ng(H )) < G and |Ng (NG(H ))| = p3 where p is aprime. By a corollary of our main result we obtain some inequality which corresponds to the fact
|G : Ng(Ng(H))| < p + 1.
Keywords: Association schemes, p-schemes. Math. Subj. Class. (2020): 05E15, 05E30
1 Introduction
Let G be a finite group with a subgroup H which satisfies
H<Ng(H) <Ng(Ng(H)) < G and |NG(NG(H))| = p3	(1.1)
where p is a prime. In this article we focus on association schemes axiom-zing some properties derived from the orbitals of the action of G on G/H.
We shall recall some terminologies to show that the definition of coherent configurations is derived from properties of the binary relations obtained from a permutation group. Let G be a permutation group of a finite set Q. Then G acts on Q x Q by its entry-wise action, i.e.,
(a,/)x :=(ax,/x) for a,/ £ Q and x £ G.
We denote the set of orbits of the action of G on Q x Q by Inv(G), which satisfies the following conditions:
*This work was supported by BK21 Dynamic Math Center, Department of Mathematics at Pusan National University.
Î Corresponding author.
E-mail addresses: wasimabbas@pusan.ac.kr (Wasim Abbas), hirasaka@pusan.ac.kr (Mitsugu Hirasaka)
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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Ars Math. Contemp. 19 (2020) 52-23
(i)	The diagonal relation 1q is a union of elements of Inv(G);
(ii)	For each s G Inv(G) we have s* G Inv(G) where s* := {(a, ,0) | (3, a) G s};
(iii)	For all s,t, u G Inv(G) we have asat = J2ueS cUUia« for cUt G N uniquely determined by s,t,u where au is the adjacency matrix of u,i.e., (ffu)a„s = 1 if (a,,0) G u and (&u)a.,ft = 0 if (a, ,0) G u.
A coherent configuration is a pair (Q, S) of a finite set Q and a partition S of Q x Q which satisfies the conditions obtained from the above by replacing Inv(G) by S. We say that a coherent configuration (Q, S) is schurian if S = Inv(G) for some permutation group G of Q, and it is homogeneous or an association scheme if 1q g S (see [2] and [3] for its background).
Suppose that G has a subgroup H which satisfies (1.1). Then |H| = p, |NG(H)| = p2 and for each g g G we have the following:
(i)	|HgH|/|H| G {l,p} and |NG(H)gNG(H)|/|NG(H)| G {l,p};
(ii)	|HgH|/|H| = l if and only if g G Ng(H);
(iii)	|Ng (H )gNG(H )|/|Ng (H )| = l if and only if g G Ng(Ng(H));
(iv)	Ng(Ng(H)) is the smallest normal subgroup of G containing H.
Since G acts faithfully and transitively on the set of right cosets of H in G by its right multiplication, it induces a schurian association scheme (Q, S) where Q = {Hx | x G G} and S = Inv(G) such that, for each s G S we have the following:
(i)	ns G {l,p} where ns := c^*;
(ii)	Oe(S) forms a group of order p where Oe (S) := {s G S | ns = l};
(iii)	Oe(S) = {s G S | ss*s = s} where Oe(S) is the thin residue of S (see Section 2, [9] or [10] for its definition).
The following is our main result:
Theorem 1.1. Let (Q, S) be an association scheme with Oe(S) < Oe(S) such that ns G {l,p} for each s G S and nOe(S) = p2 where p is a prime. Then |Q| < p2(p + l).
In [4] they give a criterion on association schemes whose thin residue Oe (S) induces the subschemes isomorphic to either
Cp2, Cp x Cp or Cp I Cp.
Here we denote (G, Inv(G)) by G when G acts on itself by its right multiplication and we denote the wreath product of one scheme (A, U) by another scheme (r, V) by (A, U) I (r, V), i.e.,
(A, U) I (r, V) := (A x r, {lr < u | u G U} U {v <g> U | v G V \ {lr}})
where
lr < u := {((¿i, y), (¿2,7)) | (¿1, ¿2) G u, 7 G r} and v < U := {((¿1 ,yi), (¿2,72)) | ¿1, ¿2 G A, (71,72) G v}.
W Abbas and M. Hirasaka: Association schemes with a certain type of p-subschemes
53
For the case of O6(S) ~ Cp2 we can apply the main result in [7] to conclude that (Q, S) is schurian. For the case of O6 (S) ~ Cp x Cp we can say that | Q | < p2 (p2 + p + 1) under the assumption that ns = p for each s G S \ O6 (S). For the case of O6 (S) ~ Cp ^ Cp we had no progression for the last five years.
In [6] all association schemes of degree 27 are classified by computational enumeration, and there are three pairs of non-isomorphic association schemes with O6 (S) ~ C3 I C3 which are algebraic isomorphic. These examples had given an impression that we need some complicated combinatorial argument to enumerate p-schemes (Q, S) with O6(S) ~ Cp I Cp and {ns | s G S \ O6(S)} = {p}. The following reduces our argument to the p-schemes of degree p3 where an association scheme (Q, S) is called a p-scheme if |s | is a power of p for each s g S:
Corollary 1.2. For each p-scheme (Q, S) with O6(S) ~ Cp I Cp, if ns = p for each s G S \ O6(S),then |Q| = p3.
In the proof of Theorem 1.1 the theory of coherent configurations plays an important role through the thin residue extension which is a way of construction of coherent configurations from association schemes (see [5, Theorem 2.1] or [8]). The following is the kernel of our paper:
Theorem 1.3. For each coherent configuration (Q, S) whose fibers are isomorphic to Cp I Cp, if |s| = p3 for each s G S with asas = 0, then either |Q| < p2(p + 1) or ss*s = s for each s G S.
In Section 2 we prepare necessary terminologies on coherent configurations. In Section 3 we prove our main results.
2 Preliminaries
Throughout this section, we assume that (Q, S) is a coherent configuration. An element of Q and an element of S are called a point and a basis relation, respectively. Furthermore, |Q| and |S| are called the degree and rank of (Q, S), respectively. For all a,,0 G Q the unique element in S containing (a, ft) is denoted by r(a, ft). For s G S and a G Q we set
as := {£ G Q | (a,£) G s}.
A subset A of Q is called a fiber of (Q, S) if 1a G S. For each s g S, there exists a unique pair (A, r) of fibers such that s C A x r. For fibers A, r of (Q, S) we denote the set of s G S with s C A x r by SA,r, and we set Sa := Sa,a. It is easily verified that (A, Sa) is a homogeneous coherent configuration. Now we define the complex product on the power set of S as follows: For all subsets T and U of S we set
TU := {s G S | cstu > 0 for some t G T and u G U}
where the singleton {t} in the complex product is written without its parenthesis. The following equations are frequently used without any mention:
Lemma 2.1. Let (Q, S) be a coherent configuration. Then we have the following:
(i)	For all r, s G S, if rs = 0, then nrns = ^t£S ctrsnt;
(ii)	For all r, s,t G S we have |t|ct*s = |r|c^ = |s|cst;
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Ars Math. Contemp. 19 (2020) 54-23
(iii) For all r, s G S we have |{t G S | t G rs}| < gcd(nr ,ns). For T C SA,r we set
nT	nt.
teT
Here we mention closed subsets, their subschemes and factor scheme according to the terminologies given in [10]. Let (Q, S) be an association scheme and T C S. We say that a non-empty subset T of S is closed if TT* C T where
T* := {t* | t G T},
equivalently |JteT t is an equivalence relation on Q whose equivalence classes are
{aT | a G Q}
where aT := {P G Q | (a, P) G t for some t G T}. Let T be a closed subset of S and a G Q. It is well-known (see [9]) that
(Q, S)aT := (aT, {t n (aT x aT) | t G T})
is an association scheme, called the subscheme of (Q, S) induced by aT, and that
(Q,S)t := (Q/T, S//T)
is also an association scheme where
Q/T := {aT | a G Q}, S//T = {sT | s G S} and sT := {(aT, PT) | (y, S) g s for some (y, S) g aT x PT},
which is called the factor scheme of (Q, S) over T.
We say that a closed subset T is thin if nt = 1 for each t g T, and O^ (S) is called the thin radical of S, and the smallest closed subset T such that S//T is thin is called the thin residue of S, which is denoted by Oe (S).
3 Proof of the main theorem
Let (Q, S) be a coherent configuration whose distinct fibers are Qi, Q2,..., Qm. For all integers i, j with 1 < i,j < m we set
Sij := ,Qj and Si := S
Throughout this section we assume that (Qi,Si) ~ Cp I Cp for i = 1,2,... ,m where p is a prime and Cp I Cp is a unique non-thin p-scheme of degree p2 up to isomorphism.
For s G S we say that s is regular if ss*s = {s} and we denote by R the set of regular elements in S.
Lemma 3.1. For each regular element s G Sij with ns = p we have
(?s* = p(Eteo0(si) at) and as*as = p(Eteo0(S,) at). In particular, ss* = Og(Si) and s* s = Og (Sj).
W Abbas and M. Hirasaka: Association schemes with a certain type of p-subschemes
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Proof. Notice that {1^} C ss* c S\ and ts = {s} for each t e ss*. Since {t e Si | ts = {s}} is a closed subset of valency at most ns, it follows from (Qi, Si) ~ Cp I Cp that ss* = Og(Si), and hence for each t e ss*
cL* = cltns* /nt* = p.
This implies that asas* = p(J2teOf)(S¿) at). By the symmetric argument we have as* as =
P(Eieo0(S,) at).	□
Lemma 3.2. For each non-regular element s G Sij with ns = p we have
asas* = p^i + E„eSi\O0 (Si) au and °s* = P^Ul, + YsueSj\Oe (Sj ) au.
Proof. Notice that {t G Si | ts = {s}} = }, otherwise, s is regular or ns = p2, a contradiction. This implies that the singletons ts with t G Og(Si) are distinct elements of valency p. Since
p2 = |Qj| =^2 ns >	nts = TP + P +-----+ P = P2,
seSij teog (Si)
it follows that Og(sSi)s = Sij.
We claim that Si \ Og(Si) C ss*. Let u G Si \ Og(Si). Then there exists t G Og(Si) such that u G tss* since u G Sijs* = Og(Si)ss*. This implies that u = t*u C t*(tss*) = ss*.
By the claim with p2 = nsns» = ^teS. css*tnt and css»i^ = ns = p we have the first statement, and the second statement is obtained by the symmetric argument. □
For the remainder of this section we assume that ns = p for each s G |Ji=j Sij.
Lemma 3.3. The set UseR s is an equivalence relation on Q.
Proof. Since G Si C R for i = 1,2,..., m, |JseR s is reflexive. Since ss*s = {s} is equivalent to s*ss* = {s*}, UseR s is symmetric.
Let a G Qi, ¡ G Qj and y G Qk with r(a, ¡), r(¡, j) G R. Then we have
r(a,Y)r(a,Y)* C r(a, ¡)r(3,j)r(3,j)*r(a, ¡3)*.
If one of r(a, ß), r(ß, 7) is thin, then (a, j)r(a, 7)*, and hence r(a, j) G R. Now we assume that both of them are non-thin. Since r(ß,j)r(ß,j)* = Og (Sj) = r(a, ß)*r(a, ß), it follows that
r(a,Y)r(a,Y)* C r(a, ß)r(a, ß)* = Og(Si).
Applying Lemma 3.1 and 3.2 we obtain that r(a, j) is regular, and hence |JseR s is transitive.	□
Lemma 3.4. The set |JseN s is an equivalence relation on Q where N := |JSi U
(S \ R).
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Ars Math. Contemp. 19 (2020) 56-23
Proof. Since € Si Ç N for i = 1,2,..., m, |JseN s is reflexive. By Lemma 3.3, UseR is symmetric, so that UseN s is symmetric.
Let a € Qi, ß € Qj and y € Qk with r(a, ß), r(ß, y) € N. Since USi Ç R, it follows from Lemma 3.3 that it suffices to show that
r(a,Y) € S \ R
under the assumption that
r(a,ß),r(ß,Y) € S \ R with i = k. Suppose the contrary, i.e., r(a, y) € R. Then, by Lemma 3.3, Sik Ç R. Since
r(a,ß)r(ß,Y) Ç Sik Ç R,
it follows that
Og(Si)r(a, ß)r(ß, y) = r(a, ß)r(ß, y). On the other hand, we have
Og (Si)r(a,ß)r(ß,Y) = Sj r(ß,Y) = Sik.
Thus, r(a, ß)r(ß, y) = Sik. Since i = k, each element of Sik has valency p, and hence,
aSl = ^ au
uesik
where s1 := r(a, ß) and s2 := r(ß, y). By Lemma 3.2,
P2 = {osi,^si) = {°*S1 °si<2) = P2 + P(P - 1), a contradiction where { , ) is the inner product defined by
{A,B) := 1/p2tr(AB*) for all A,B € Mn(C). Therefore, UseN s is transitive.	□
Lemma 3.5. We have either R = S or N = S.
Proof. Suppose R = S. Let a, ß € Q with r(a, ß) € R. Since R = S, there exists y € Q with r(a, y) € N. Notice that r(ß, y) € R U N .By Lemma 3.3, r(ß, y) € N, and hence, by Lemma 3.4,
m
r(a,ß) € R n N = y Si.
i=i
Since a, ß € Q are arbitrarily taken, it follows that
m
R = y Si and N = S.	□
i=i
Lemma 3.6. Suppose that S = N and s1 € Sij, s2 € Sjk and s3 € Sik with distinct i,j,k. Then oSl oS2 = oS3 teOs(Sfc) atot) for some non-negative integers at
with T,teoe(sk) at = P> T,teoe(sk) a2 = 2P - 1 and for each u € Og(Sk) \ {1Qfc}
^teo0(Sk) atatu = P - 1.
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Proof. Since S1S2 Q Sj = s3Og(Sk), aSl&s2 = J2teoe(sk) atas31 for some nonnegative integers at. Since a&31 = a&3at and
p2 = nSi nS2 = ^ atnS3t = p ^ at,
t£Oe(Sk)	tOe(Sj)
it remains to show the last two equalities on at with t G Og(Sj). Expanding a*&2a*Si aSla&2 by two ways we obtain from Lemma 3.2 that
(2p2 - p)aln. + (p2 - p) Y, at + (P2 - 2P) Y,
teo0 (S3 )\{in.}	ueSj\oe(Sj )
teoe (Sk)	teoe (Sk)
Therefore, we conclude from Lemma 3.2 that
P	a"2 = 2p2 - p and p ^ a4atu = p2 - P
teoe(Sk)	teoe(Sk)
for each u G Og (Sk) with u = .	□
For the remainder of this section we assume that
S = N.
For i = 1,2,... ,m we take ai G Q. and we define ti G Si such that t1 G Og (S1)\{1n1}, and for i = 2, 3,... ,m, ti is a unique element in Og (Si) with r(a1t1,aiti) = r(a1, ai). Then Cp acts semi-regularly on Q such that
Q x Cp ^ Q, (¡3i,tj) ^ ¡34, where Cp = (t) and 3i is an arbitrary element in Qi.
Lemma 3.7. The above action acts semi-regularly on Q as an automorphism of (Q, S).
Proof. Since Cp acts regularly on each of geometric coset of Og(Sj,) for i = 1,2,..., m, the action is semi-regular on Q. By the definition of {ti}, it is straightforward to show that r(a1,ai) is fixed by the action on Q x Q, and hence each element of |Jm=2 S1j U Sj1 is also fixed since S1j = Og(S1)r(a1,aj). Let s G Sijwith 2 < i,j. Notice that r(ai, a1)r(a1, a.j) is a proper subset of Sij by Lemma 3.6. This implies that s is obtained as the intersection of some of tkr(ai,a1)r(a1,aj) with 0 < k < p - 1, and hence s is fixed.	□
For each i = 1,2,... ,m we take {aik | k = 1,2,..., m} to be a complete set of representatives with respect to the equivalence relation |Jt£oe(S,) t on Qi.
Lemma 3.8. For each s G Sj with i = j and all k,l = 1,2,...,p there exists a unique h(s)ki G Zp such that r(aik, ajlth(s)kl) = s. Moreover, if s1 G Sj and ta G Og(Sk) with s1 = sta, then h(s1 )kl = h(s)kl + a for all k,l = 1, 2,..., m.
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Ars Math. Contemp. 19 (2020) 58-23
Proof. Since Og (Sj) acts regularly on Sij by its right multiplication, the first statement holds. The second statement is obtained by a direct computation.	□
Lemma 3.9. For each s G Sij with i = j and all k,l = 1,2,... ,p we have
s n (aik Og (Si) x ajlOg (Sj)) = {(aikt1, ajt tj) | b — a = h(s)kl}. Proof. Notice that
r(aiktai, a.jitj) = (t?)*r(aik,aji)tbj = r(aik,aji)tj-a.
Since r(aik ,a.ji th(s)kl) = s by Lemma 3.8, it follows that r(aik ti, ajitij) = s if and only if b — a = h(s)ki.	□
Proposition 3.10. For each s G Sij with i = j the matrix (h(s)kl) G Mpxp(Zp) satisfies that, for all distinct ki,k2 G {1, 2,... ,p},
{h(s)kui — h(s)k2,i | l = 1, 2,.. . ,p} = Zp.
In other word the matrix is a generalized Hadamard matrix of degree p over Zp, equiv-alently, the matrix (£h(s)kl) g Mpxp(C) is a complex Hadamard matrix ofButson type (p,p) where £ is a primitive p-th root of unity.
Proof. Notice that, for all distinct k, l, by Lemma 3.9,
{y G Q | r(aikti,Y)= r(autbi,1) = s}
equals
p
[J{ajrj | c — a = h(s)kr,c — b = h(s)w}.
r=i
Since the upper one is a singleton by Lemma 3.2, there exists a unique r G {1,2,... ,p} such that b — a = h(s)kr — h(s)ir. Since a and b are arbitrarily taken, the first statement holds.
The second statement holds since J2p=—0 X is the minimal polynomial of £ over Q. □
We shall write the matrix (£h(s)kl) as H(s). For s G Sij with i = j, the restriction of as to Qi x Qj can be viewed as a (p x p)-matrix whose (k, l)-entry is the matrix ph(s)kl where Pi is the permutation matrix corresponding to the mapping ^ ftiti where we may assume that Pi = Pj, say P, for all i,j = 1, 2,... ,m by Lemma 3.7. Notice that H (s) is obtained from (Ph(s)kl) by sending Ph(s)kl to £h(s)kl.
Proposition 3.11. For all si G Sij, s2 G Sjk and s3 G Sik with distinct i,j, k we have H(si)H(s2) = a.H(s3) for some a G C with |a| = fp.
Proof. By Lemma 3.6, H(si)H(s2) = H(s3)^p=roi ai£i) for some a-i G Z where a-i = ct^. Thus, it suffices to show that p—0 a,i£i) |2 = p. By Lemma 3.6, the left hand side
eqkuals
p- i p- i p- i p- i p-i
££ aiaj£i-j = £ a2 + ££ aja+j£i = (2p — 1) + (p — 1)( — 1)= p. □
i=0 j—o	i=0 i=i j—o
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Corollary 3.12. Let si := r(a1, ai) for i = 2, 3,..., m and Bi denote the basis consisting of the rows of H(si), i = 2,3,..., m, and B1 be the standard basis. Then {B1, B2,..., Bm} is a mutually unbiased bases for Cp, and m < p + 1.
Proof. The first statement is an immediate consequence of Proposition 3.10, and the second statement follows from a well-known fact that the number of mutually unbiased bases for Cn is at most n + 1 (see [1]).	□
Proof of Theorem 1.3. Suppose that R = S. Then N = S and the theorem follows from Corollary 3.12.	□
Proof of Theorem 1.1. Since nOe (S) = p2 and Oe (S ) < Oe (S ), it follows from [ , Theorem 2.1] (or see [8]) that the thin residue extension of (Q, S) is a coherent configuration with all fibers isomorphic to Cp i Cp such that each basic relation out of the fibers has valency p.
We claim that S = N. Otherwise, S = R, which implies that (ss* | s G S} has valency p. Since Oe(S) = (ss* | s G S} (see [9]), it contradicts that Oe(S) has valency p2.
By the claim, S = N. Since the number of fibers of the thin residue extension of (Q, S) equals |Q/Oe(S)|, the theorem follows from Theorem 1.3.	□
Proof of Corollary 1.2. Since (Q, S ) is a p-scheme and Oe (S) ~ Cp x Cp, |Q| is a power of p greater than p2. By Theorem 1.1, |Q| < (p + 1)p2, and hence, |Q| = p3.	□
ORCID iDs
Wasim Abbas © https://orcid.org/0000-0002-1706-1462 References
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