Adv. Geom. 4 (2004), 241–261
Advances in Geometry
( de Gruyter 2004
Association schemes of a‰ne type over finite rings
Hajime Tanaka*
(Communicated by E. Bannai)
Abstract. Kwok [10] studied the association schemes obtained by the action of the semidirect
products of the orthogonal groups over the finite fields and the underlying vector spaces. They
are called the assiciation schemes of a‰ne type. In this paper, we define the association
schemes of a‰ne type over the finite ring Zq ¼ Z=qZ where q is a prime power in the same
manner, and calculate their character tables explicitly, using the method in Medrano et al. [13]
and DeDeo [8]. In particular, it turns out that the character tables are described in terms of the
Kloosterman sums. We also show that these association schemes are self-dual.
Introduction
The purpose of the present paper is to study a certain kind of association schemes
related to the orthogonal groups over the finite ring Zq ¼ Z=qZ, where q ¼ p r is a
prime power.
Let f : Fqn ! Fq be a non-degenerate quadratic form over the finite field Fq and
OðFqn ; f Þ its orthogonal group. Since id is contained in OðFqn ; f Þ, the action of the
semidirect product OðFqn ; f Þ y Fqn on Fqn defines a symmetric association scheme
XðOðFqn ; f Þ; Fqn Þ. Kwok [10] called this association scheme an association scheme of
a‰ne type, and calculated its character table completely (see also [12], [5]).
We define the association schemes of a‰ne type over the finite rings Zk ¼ Z=
kZ ðk A NÞ in the same manner. However, by the Chinese remainder theorem, it is
enough to consider the case where k is a prime power ([1, p. 59]). It seems that these
association schemes had not been studied, but some related results can be found in
Medrano et al. [13] and DeDeo [8]. Namely, in [13], [8], the finite Euclidean graph
n
Xq ðn; aÞ over Zq with a A Z
q ¼ Zq n pZq is defined as the graph with the vertex set Zq ,
and the edge set
E ¼ fðx; yÞ A Zqn Zqn j dðx; yÞ ¼ ag;
* The author is supported in part by a grant from the Japan Society for the Promotion of
Science.
242
Hajime Tanaka
where dðx; yÞ A Zq is the ‘‘distance’’ defined by
dðx; yÞ ¼ ðx1 y1 Þ 2 þ ðx2 y2 Þ 2 þ þ ðxn yn Þ 2 :
We will show that if q is an odd prime power, then in our language these graphs
are part of the relations of XðOðZqn ; dð ; 0ÞÞ; Zqn Þ, the association scheme of a‰ne
type with respect to the non-degenerate quadratic form dðx; 0Þ ¼ x12 þ x22 þ þ xn2
(dð ; 0Þ is degenerate if q is even).
In this paper, we determine the character table of the symmetric association
scheme XðOðZqn ; f Þ; Zqn Þ explicitly for all non-degenerate quadratic forms on Zqn (for
both odd q and even q). These results are given in Theorem 2.10, Theorem 2.12 and
Theorem 2.15. In particular, we will be able to see a phenomenon similar to the Ennola type dualities observed in [4]. Also, as an immediate consequence of these calculations, we verify that these association schemes are self-dual.
The outline of the paper is as follows. In Section 1, we review some basic notions
on commutative association schemes, and classify the non-degenerate quadratic
forms on Zqn completely. In Section 2, the character tables are calculated explicitly.
The discussion in this section is almost parallel to those in [13], [8]. We will find that
the method of computing the eigenvalues of the graphs Xq ðn; aÞ used successfully in
[13], [8] also works in our case. (It seems possible to obtain the results in [10], by the
method in [12], [13], [8].) In particular, the character tables are described in terms of
the Kloosterman sums over Zq .
Acknowledgement. The author would like to thank Mr. Makoto Tagami for some
useful discussions.
1
Preliminaries and the classification of the non-degenerate
quadratic forms over Zq
1.1 Preliminaries on commutative association schemes. Here, we recall some basic
notions on commutative association schemes. We refer the reader to [3], [7], [2] for
the background in the theory of these objects.
Let X ¼ ðX ; fRi g0cicd Þ be a commutative association scheme with the adjacency
matrices A0 ¼ I ; A1 ; . . . ; Ad , where I is the identity matrix of degree jxj. The algebra
A of dimension d þ 1, generated by A0 ; . . . ; Ad over the complex number field C, is
called the Bose–Mesner algebra of X. If we consider the action of A on the vector
space V ¼ CjX j indexed by the elements of X , then V is decomposed into the direct
sum of the maximal common eigenspaces:
V ¼ V0 ? V1 ? ? Vd ;
where V0 is the one-dimensional subspace spanned by the all-one vector. Let
Ei : V ! Vi be the orthogonal projection ð0 c i c dÞ. Then the set fE0 ; E1 ; . . . ; Ed g
forms another basis of A, and the base change matrix P ¼ ðpi ð jÞÞ is called the character table or the first eigenmatrix of X:
Association schemes of a‰ne type over finite rings
Ai ¼
d
X
pi ð jÞEj
243
ð0 c i c dÞ
j¼0
(the ð j; iÞ-entry of P is pi ð jÞ). In particular, ki ¼ pi ð0Þ is the valency of the regular
graph ðX ; Ri Þ. The second eigenmatrix Q ¼ ðqi ð jÞÞ of X is defined by Q ¼ jX jP1 ,
that is,
jX jEi ¼
d
X
qi ð jÞAj
ð0 c i c dÞ:
j¼0
The numbers mi ¼ qi ð0Þ ¼ dim Vi ð0 c i c dÞ are called the multiplicities of X. Notice that the first eigenmatrix P, together with the multiplicities of X, gives complete
information of the spectra of the graphs ðX ; Ri Þ ð1 c i c dÞ.
Now, assume that X is symmetric and that the underlying set X has the structure of
an abelian group. We call X a translation association scheme if for 0 c i c d and
z A X we have
ðx; yÞ A Ri ) ðx þ z; y þ zÞ A Ri :
For such an association scheme, there is a natural way to define the dual scheme
X ¼ ðX ; fRi g0cicd Þ, where X denotes the character group of X . Namely, we
define the relation Ri by
ðm; nÞ A Ri , nm1 A Vi
(considered as a vector of V ). Then, X ¼ ðX ; fRi g0cicd Þ becomes a translation
association scheme with the eigenmatrices P ¼ Q and Q ¼ P (see e.g. [7, §2.10B] or
[3, §2.6]). The translation association scheme X is called self-dual if it is isomorphic to
its dual X . In particular, if X is self-dual, then clearly we have P ¼ Q.
1.2 The classification of the non-degenerate quadratic forms over Zq . Let q ¼ p r with
p prime. If a is a unit in Zq ¼ Z=qZ, we denote its multiplicative inverse in Zq by
a½1 . Sometimes we identify the ring Zq with the set f0; 1; . . . ; q 1g, and regard
Zpnl ðl < rÞ as a subset of Zqn .
For a nonzero element a of Zq , we denote the largest integer l such that p l divides
ðrÞ
a by ordðrÞ
p ðaÞ. Conventionally, ordp ð0Þ is defined to be r. Also, if x ¼ ðx1 ; x2 ; . . . ; xn Þ
n
is an element of Zq , then we define ordðrÞ
p ðxÞ by
ðrÞ
ordðrÞ
p ðxÞ ¼ min ordp ðxi Þ:
1cicn
The reduction of x modulo pZqn is denoted by x A Zpn . If ordðrÞ
p ðxÞ > 0, then there
exists a unique element y of Zpnr1 such that x ¼ py, and we write y ¼ 1p x.
For later use, we prove the following lemma.
244
Hajime Tanaka
Lemma 1.1. Let W be a submodule of Zqn . Then W is a direct summand of Zqn if and
only if it is free.
Proof. First, suppose that W is a direct summand of Zqn so that W is projective. Then
since Zq is local with the maximal ideal pZq , W is free (see e.g. [11, p. 9, Theorem
2.5]).
Conversely, suppose that W is free. Let f f1 ; f2 ; . . . ; fk g be a basis of W . Then it
follows that f 1 ; f 2 ; . . . ; f k A Zpn are linearly independent over Zp . In fact, assume
that a1 f 1 þ a2 f 2 þ þ ak f k ¼ 0 holds for some a1 ; a2 ; . . . ; ak A Zq . Then we have
f ¼
X
ai fi A pZqn :
paai
This implies f ¼ 0, since otherwise p r1 f does not vanish, which is a contradiction.
Thus, we have ai ¼ 0 for all i, as desired. Now, take fkþ1 ; . . . ; fn A Zqn so that
f f 1 ; f 2 ; . . . ; f n g forms a basis of Zpn . Then by the well-known NakayamaPlemma (see
n
e.g. [11, p. 8, Theorem 2.2]), f1 ; f2 ; . . . ; fn span Zqn . Finally, since j i¼1
Z q fi j ¼
jZqn j ¼ q n , f1 ; f2 ; . . . ; fn must be linearly independent over Zq . This completes the
proof of Lemma 1.1.
r
Corollary 1.2. Let x be an element of Zqn . Then Zq x is a direct summand of Zqn if and
only if ordðrÞ
p ðxÞ ¼ 0.
Let B : Zqn Zqn ! Zq be a symmetric bilinear form. For a subset U of Zqn , we
define the orthogonal complement U ? of U by
U ? ¼ fx A Zqn j Bðx; yÞ ¼ 0 for all y A Ug:
The symmetric bilinear form B is said to be non-degenerate if detðBðei ; ej ÞÞ A Z
q ¼
Zq n pZq , where fe1 ; e2 ; . . . ; en g is the standard basis of Zqn , that is, ei is the element
of Zqn with 1 in the i-th component and zero elsewhere. Clearly, this condition is
equivalent to saying that the map x A Zqn 7! Bðx; Þ A HomZq ðZqn ; Zq Þ gives an isomorphism between Zqn and HomZq ðZqn ; Zq Þ.
A quadratic form on Zqn is a map f : Zqn ! Zq satisfying
f ðaxÞ ¼ a 2 f ðxÞ;
f ðx þ yÞ ¼ f ðxÞ þ f ð yÞ þ Bf ðx; yÞ;
for any a A Zq and x; y A Zqn , where Bf is a symmetric bilinear form on Zqn . Sometimes we call the pair ðZqn ; f Þ a quadratic module over Zq .
Let ðZqm ; f 0 Þ be another quadratic module over Zq . An isometry s : ðZqn ; f Þ !
ðZqm ; f 0 Þ is an injective Zq -linear map such that f ðxÞ ¼ f 0 ðsðxÞÞ for all x A Zqn and
sðZqn Þ is a direct summand of Zqm . If in addition the isometry s is a linear isomor-
Association schemes of a‰ne type over finite rings
245
phism, then we say that the two quadratic modules ðZqn ; f Þ and ðZqm ; f 0 Þ are isomorphic, and write ðZqn ; f Þ G ðZqm ; f 0 Þ.
The quadratic form f is said to be non-degenerate if Bf is non-degenerate. We define the reductions f : Zpn ! Zp and Bf : Zp Zp ! Zp of f and Bf modulo pZq , in
an obvious manner.
The orthogonal group OðZqn ; f Þ is the group of all linear transformations on Zqn
that fix f , that is,
OðZqn ; f Þ ¼ fs A GLðZqn Þ j f ðsðxÞÞ ¼ f ðxÞ for all x A Zqn g:
Now we classify all non-degenerate quadratic forms on Zqn using the classification
of those on Zpn . First of all, we prepare two propositions.
Proposition 1.3 (cf. [1, p. 10, Proposition 3.2]). Let f be a quadratic form on Zqn . If W
is a direct summand of Zqn such that the restriction f jW of f to W is non-degenerate,
then we have Zqn ¼ W ? W ? .
Proof. For an element x of Zqn , define a Zq -linear map jx : W ! Zq by jx ðyÞ ¼
Bf ðx; yÞ for y A W . Since Bf jW is non-degenerate, there exists a unique element z
of W such that jx ð yÞ ¼ Bf ðz; yÞ for all y A W , so that we have x ¼ z þ ðx zÞ A
W þ W ? . Since clearly W V W ? ¼ 0, we obtain the desired result.
r
Proposition 1.4 (cf. [1, p. 11, Corollary 3.3]). Let f be a quadratic form on Zqn .
Then, for any orthogonal decomposition Zpn ¼ W 1 ? W 2 with respect to f such that
the restriction f jW1 is non-degenerate, there exists an orthogonal decomposition
Zqn ¼ W1 ? W2 such that f jW1 is non-degenerate and Wi G Wi =pWi ði ¼ 1; 2Þ.
Proof. Let y1 ; y2 ; . . . ; yn be a basis of Zpn such that y1 ; y2 ; . . . ; yl span W1 . For
each yi ¼ ðyi1 ; yi2 ; . . . ; yin Þ ð1 c i c nÞ, take an element xi ¼ ðxi1 ; xi2 ; . . . ; xin Þ of Zqn
such that yi ¼ xi . Since detð yij Þ1ci; jcn 0 0, we have detðxij Þ1ci; jcn A Z
q so that
x1 ; x2 ; . . . ; xn form a basis of Zqn . Put
W1 ¼ Zq x1 l Zq x2 l l Zq xl :
Since f jW1 is non-degenerate, we have detðBf ðyi ; yj ÞÞ1ci; jcl 0 0, from which it follows that detðBf ðxi ; xj ÞÞ1ci; jcl A Z
q , that is, f jW1 is non-degenerate. Thus, we have
Zqn ¼ W1 ? W2 by Proposition 1.3 where W2 ¼ W1? and clearly, Wi G Wi =pWi ði ¼
1; 2Þ, as desired.
r
It is well-known that the non-degenerate quadratic forms over Zp are classified as
follows:
Theorem 1.5 (cf. [14]). (i) Suppose n ¼ 2m is even. If p is odd, then there are two inequivalent non-degenerate quadratic forms f1þ and f1 :
246
Hajime Tanaka
f1þ ðxÞ ¼ x1 x2 þ þ x2m1 x2m ;
2
2
ex2m
f1 ðxÞ ¼ x1 x2 þ þ x2m3 x2m2 þ x2m1
for x ¼ ðx1 ; x2 ; . . . ; x2m Þ A Zp2m , where e is a non-square element of Zp .
If p ¼ 2, then there are also two inequivalent non-degenerate quadratic forms f1þ and
f1 :
f1þ ðxÞ ¼ x1 x2 þ þ x2m1 x2m ;
2
2
þ x2m1 x2m þ x2m
f1 ðxÞ ¼ x1 x2 þ þ x2m3 x2m2 þ x2m1
for x ¼ ðx1 ; x2 ; . . . ; x2m Þ A Z22m .
(ii) Suppose n ¼ 2m þ 1 is odd. If p is odd, then there are two inequivalent nondegenerate quadratic forms f1 and f10 :
2
f1 ðxÞ ¼ x1 x2 þ þ x2m1 x2m þ x2mþ1
;
2
f10 ðxÞ ¼ x1 x2 þ þ x2m1 x2m þ ex2mþ1
for x ¼ ðx1 ; x2 ; . . . ; x2mþ1 Þ A Zp2mþ1 , where e is a non-square element of Zp , but their
orthogonal groups OðZp2mþ1 ; f1 Þ and OðZp2mþ1 ; f10 Þ are isomorphic.
If p ¼ 2, then there is no non-degenerate quadratic form on Z22mþ1 .
Remark 1.6. The definition of non-degeneracy of a quadratic form in this paper is
slightly stronger than that in [14]. Namely, if we define the radical Rad f of a quadratic form f on Zpn by
Rad f ¼ f 1 ð0Þ V ðZpn Þ? ;
then using the terminology in [14], f is said to be non-degenerate if Rad f ¼ 0. Our
definition agrees with this unless p ¼ 2 and n is odd. If one adopts the definition in
[14], then it turns out that there exists exactly one inequivalent non-degenerate quadratic form f1 on Z22mþ1 :
2
f1 ðxÞ ¼ x1 x2 þ þ x2m1 x2m þ x2mþ1
;
for x ¼ ðx1 ; x2 ; . . . ; x2mþ1 Þ A Z22mþ1 , which is clearly degenerate in our sense.
For the convenience of the discussion, we call the (free) quadratic module
ðZq e1 l Zq e2 ; f Þ of rank two with a distinguished basis fe1 ; e2 g such that f ðe1 Þ ¼ a,
f ðe2 Þ ¼ b and Bf ðe1 ; e2 Þ ¼ 1, the quadratic module of type ½a; b. Similarly, we call
the quadratic module ðZq e; f Þ of rank one with a distinguished basis feg such that
f ðeÞ ¼ z, the quadratic module of type ½z. In order to find the isomorphisms among
these quadratic modules, we need the following lemma.
Association schemes of a‰ne type over finite rings
247
Lemma 1.7. (i) If p is odd, then for any u A pZq , there exists a unique element a A Zq
such that a 1 1 mod p and a 2 a 1 u mod q.
(ii) If p ¼ 2, then for any odd t and even u in Zq , there exist unique even a and odd b
in Zq such that a 2 þ ta 1 b 2 þ tb 1 u mod q.
Proof. (i) Let a and b be elements of 1 þ pZq . Then we have a 2 a 1 b 2 b mod q if
and only if ða bÞða þ b 1Þ 1 0 mod q. Since a þ b 1 1 1 mod p, this implies
a 1 b mod q. Therefore a 2 a takes all u A pZq as a runs through 1 þ pZq .
(ii) The proof is similar to that of (i), hence omitted.
r
Using Lemma 1.7, we obtain the following.
Proposition 1.8. (i) For any a and b in pZq , the quadratic modules of type ½a; b and
½0; 0 are isomorphic.
(ii) If p ¼ 2, then the quadratic modules of type ½g; d and ½1; 1 are isomorphic for
any g and d in Z
q.
(iii) Suppose p is odd, and let e be a non-square element of Zp . Then for each z A Z
q,
the quadratic module of type ½z is isomorphic to that of type [1] or ½e, depending on
whether z is a square or not.
Proof. (i) Let fe1 ; e2 g be the distinguished basis of the quadratic module of type ½0; 0.
By Proposition 1.7 (i) and (ii), there exists unique a A Zq such that a 1 1 mod p and
a 2 a 1 ab mod q. Put
e10 ¼ ae1 þ a½1 ae2
and
e20 ¼ be1 þ e2 :
Then since
det
a
a½1 a
b
1
¼ a a½1 ab 1 1 mod p;
fe10 ; e20 g therefore forms another basis. Clearly, we have f ðe10 Þ ¼ a, f ðe20 Þ ¼ b and
Bf ðe10 ; e20 Þ ¼ 1.
(ii) Let fe1 ; e2 g be the distinguished basis of the quadratic module of type ½1; 1.
Then it follows from Proposition 1.7 (ii) that there exist a and unique even b in Zq
such that a 2 þ a 1 g 1 mod q and 3gb 2 3b 1 ð2a þ 1Þ 2 d 1 mod q. Put
e10 ¼ ae1 þ e2
and
e20 ¼ ð2a þ 1Þ½1 ð1 ab 2bÞe1 þ be2 :
Since b is even,
a
det
1
ð2a þ 1Þ½1 ð1 ab 2bÞ
b
!
248
Hajime Tanaka
is odd, so that fe10 ; e20 g is a basis. First, we have f ðe10 Þ ¼ a 2 þ a þ 1 1 g mod q. Also,
since
ð2a þ 1ÞBf ðe10 ; e20 Þ ¼ 2að1 ab 2bÞ þ ð2a þ 1Þab þ ð1 ab 2bÞ þ 2ð2a þ 1Þb
¼ 2a þ 1;
we have Bf ðe10 ; e20 Þ ¼ 1. Finally, it follows that
ð2a þ 1Þ 2 f ðe20 Þ ¼ ð1 ab 2bÞ 2 þ ð2a þ 1Þð1 ab 2bÞb þ ð2a þ 1Þ 2 b 2
¼ 3ða 2 þ a þ 1Þb 2 3b þ 1
1 3gb 2 3b þ 1 mod q
1 ð2a þ 1Þ 2 d mod q;
so that f ðe20 Þ ¼ d.
(iii) This is trivial, since Z
q is a cyclic group if p is odd.
r
Combining Theorem 1.5, Proposition 1.4 and Proposition 1.8, we conclude that:
Theorem 1.9. (i) Suppose n ¼ 2m is even. If p is odd, then there are two inequivalent
non-degenerate quadratic forms frþ and fr on Zq2m :
frþ ðxÞ ¼ x1 x2 þ þ x2m1 x2m ;
2
2
ex2m
fr ðxÞ ¼ x1 x2 þ þ x2m3 x2m2 þ x2m1
for x ¼ ðx1 ; x2 ; . . . ; x2m Þ A Zq2m , where e is a non-square element of Zp .
If p ¼ 2, then there are also two inequivalent non-degenerate quadratic forms frþ and
fr on Zq2m :
frþ ðxÞ ¼ x1 x2 þ þ x2m1 x2m ;
2
2
þ x2m1 x2m þ x2m
fr ðxÞ ¼ x1 x2 þ þ x2m3 x2m2 þ x2m1
for x ¼ ðx1 ; x2 ; . . . ; x2m Þ A Zq2m .
(ii) Suppose n ¼ 2m þ 1 is odd. If p is odd, then there are two inequivalent nondegenerate quadratic forms fr and fr0 on Zq2mþ1 :
2
;
fr ðxÞ ¼ x1 x2 þ þ x2m1 x2m þ x2mþ1
2
fr0 ðxÞ ¼ x1 x2 þ þ x2m1 x2m þ ex2mþ1
for x ¼ ðx1 ; x2 ; . . . ; x2mþ1 Þ A Zq2mþ1 , where e is a non-square element of Zp .
If p ¼ 2, then there is no non-degenerate quadratic form on Zq2mþ1 .
Association schemes of a‰ne type over finite rings
249
In what follows, we denote the orthogonal groups of frþ , fr and fr by GOþ
2m ðZq Þ,
GO
ðZ
Þ
and
GO
ðZ
Þ,
respectively.
It
is
easy
to
see
that
the
orthogonal
group
q
2mþ1
q
2m
2mþ1
0
; fr Þ is isomorphic to GO2mþ1 ðZq Þ.
OðZq
2
The character tables
2.1 The relations of
f ), Znq ). As in the introduction, let XðOðZqn ; f Þ; Zqn Þ
denote the symmetric association scheme obtained from the action of OðZqn ; f Þ y Zqn
on Zqn . That is, the relations of XðOðZqn ; f Þ; Zqn Þ are the orbits of OðZqn ; f Þ y Zqn in its
natural action on Zqn Zqn . Since the stabilizer of an element of Zqn in OðZqn ; f Þ y Zqn
is isomorphic to OðZqn ; f Þ, the relations of XðOðZqn ; f Þ; Zqn Þ are in one-to-one correspondence with the orbits of OðZqn ; f Þ on Zqn .
For quadratic modules over fields, there is a very famous theorem known as Witt’s
extension theorem (see e.g. [14]). Knebusch [9] proved that a similar result also holds
for quadratic modules over any local ring. In our case, this result is stated as follows.
X(O(Znq ,
Theorem 2.1 ([9, Satz 5.1]). Let f be a non-degenerate quadratic form on Zqn . Suppose
that W is a direct summand of Zqn and t : W ! Zqn is an isometry. Then there exists an
extension s A OðZqn ; f Þ of t, i.e. sjW ¼ t.
We use Theorem 2.1 to determine the relations of XðOðZqn ; f Þ; Zqn Þ.
Lemma 2.2. Let f be a non-degenerate quadratic form on Zqn . Then for any element x
n
of Zqn such that ordðrÞ
p ðxÞ ¼ 0 and a A Zp l ð0 c l < rÞ, there exists an element y of Zq
rl n
rl
such that x 1 y mod p Zq and f ðyÞ ¼ f ðxÞ þ p a.
Proof. Since Bf is non-degenerate and x 0 0, there exists an element z A Zqn such that
Bf ðx; zÞ 0 0, or equivalently, Bf ðx; zÞ A Z
q . Notice that for any b; c A Zp l we have
bBf ðx; zÞ þ p rl b 2 f ðzÞ 1 cBf ðx; zÞ þ p rl c 2 f ðzÞ mod p l
if and only if
ðb cÞfBf ðx; zÞ þ p rl ðb þ cÞf ðzÞg 1 0 mod p l :
Since Bf ðx; zÞ þ p rl ðb þ cÞ f ðzÞ D 0 mod p, this implies b ¼ c. Therefore, there exists
a unique element c of Zp l such that cBf ðx; zÞ þ p rl c 2 f ðzÞ 1 a mod p l . Clearly, y ¼
x þ p rl cz A Zqn is a desired element.
r
Remark 2.3. In fact, with the notation of Lemma 2.2, the above proof shows the
existence of an element z A Zqn such that for all v A Zqn with v 1 x mod p rl Zqn ,
f ðv þ p rl czÞ ðc A Zp l Þ are distinct and
f f ðv þ p rl czÞ j c A Zp l g ¼ f ðxÞ þ p rl Zq :
250
Hajime Tanaka
Proposition 2.4. Let f be a non-degenerate quadratic form on Zqn . Then for two nonn
zero elements x and y of Zqn , there exists an automorphism
s A OðZ
q ; f Þ such that
ðrÞ
1
1
rl
sðxÞ ¼ y if and only if ordðrÞ
where
p ðxÞ ¼ ordp ðyÞ and f p l x 1 f p l y mod p
ðrÞ
l ¼ ordp ðxÞ.
ðrÞ
ðrÞ
Proof.
The ‘‘only
if ’’ part is obvious. Assume l ¼ ordp ðxÞ ¼ ordp ð yÞ ð< rÞ and
n
f p1l x 1 f p1l y mod p rl . Then by Lemma
2.2, there exists an element u of Zq
such that u 1 p1l x mod p rl Zpnl and f ðuÞ ¼ f p1l y . Since Zq u and Zq p1l y are direct
summands of Zqn by Corollary 1.2, it follows from Theorem 2.1 that there exists
s A OðZqn ; f Þ such that sðuÞ ¼ p1l y. Then we have sðxÞ ¼ p l sðuÞ ¼ y.
r
2m
In the next subsection, we calculate the character table of XðGO
2m ðZq Þ; Zq Þ. The
discussions are almost parallel to those in Medrano et al. [13] and DeDeo [8]. In §2.3
2m
2mþ1
and §2.4, the results for XðGOþ
Þ are stated
2m ðZq Þ; Zq Þ and XðGO2mþ1 ðZq Þ; Zq
without detailed proofs.
2m
2.2 The character table of X(GOC
2m (Zq ), Zq ). By Proposition 2.4, the relations of
2m
XðGO2m ðZq Þ; Zq Þ are given as follows.
ðrÞ
ðx; yÞ A R 0 , x ¼ y;
ðrÞ
ðrÞ
ðx; yÞ A Rl; a , x y A Ll; a
ðrÞ
2m
for 0 c l < r and a A Zp rl , where Ll; a is an orbit of GO
2m ðZq Þ on Zq defined by
ðrÞ
Ll; a
¼
xA
Zq2m
j ordðrÞ
p ðxÞ
¼ l;
fr
1
rl
x 1 a mod p
pl
ðrlÞ
¼ fp l u j u A L 0; a g:
ðrÞ
ðrÞ
2m
Notice that the kl; a ¼ jLl; a j are the valencies of XðGO
2m ðZq Þ; Zq Þ.
Proposition 2.5. For 0 c l < r and a A Zp rl , we have
ðrÞ
ðrÞ
kl; a ¼ jLl; a j ¼
pðrl1Þð2m1Þ p m1 ð p m þ 1Þ;
if paa;
pðrl1Þð2m1Þ ðp m1 1Þð p m þ 1Þ; if pja:
2m
In particular, XðGO
2m ðZq Þ; Zq Þ is a symmetric association scheme of class
p r þ p r1 þ þ p ¼
if m > 1, and of class
pðp r 1Þ
p1
Association schemes of a‰ne type over finite rings
251
ðp r p r1 Þ þ ð p r1 p r2 Þ þ þ ðp 1Þ ¼ p r 1
if m ¼ 1.
ðrÞ
ðrlÞ
Proof. Since jLl; a j ¼ jL 0; a j, we only have to prove the equality when l ¼ 0. If r ¼ 1,
then it is easy to see that the assertion is true (cf. [10, §3.3]). Now, suppose r > 1.
Then for an element x of Zq2m , we have ordðrÞ
p ðxÞ ¼ 0 if and only if x 0 0, from which
it follows that
jfx A Zq2m j ordðrÞ
p ðxÞ ¼ 0; fr ðxÞ 1 a mod pgj
ðr1Þ2m m1 m
p
p
ðp þ 1Þ;
if paa;
¼
pðr1Þ2m ðp m1 1Þð p m þ 1Þ; if pja.
ð1Þ
ðrÞ
By Remark 2.3, jL 0; a j is obtained by dividing the right-hand side of (1) by p r1 . r
Medrano et al. [13] and DeDeo [8] determined the graph spectra of the graphs
ðrÞ
ðR 0; a ; Zq2m Þ ða A Z
q Þ completely for many cases. We apply this method in the calcu2m
lation of the character table of XðGO
2m ðZq Þ; Zq Þ.
ðrÞ
Let V be the complex vector space of the functions j : Zq2m ! C. For each
ðrÞ
0 c l < r and a A Zp rl , we define the adjacency operator Al; a on V ðrÞ by
ðrÞ
Al; a jðxÞ ¼
X
jðyÞ
ðrÞ
xy A Ll; a
for all j A V ðrÞ . We will decompose V ðrÞ into the direct sum of the maximal common
ðrÞ
eigenspaces of the Al; a ’s.
For brevity, we denote the associated bilinear form Bfr of fr simply by B
r . Also,
we use the notation
pffiffiffiffiffiffiffi
eðrÞ ðaÞ ¼ expð2p 1a=p r Þ
ðrÞ
for a A Zq . For each u A Zq2m , define a linear character eu : Zq2m ! C by
ðrÞ
eðrÞ
u ðxÞ ¼ e ðBr ðu; xÞÞ
ðrÞ
for x A Zq2m . Then, since B
r is non-degenerate, the eu ’s are distinct and they form an
orthonormal basis of V ðrÞ with respect to the inner product
ðj; cÞ ¼
1 X
jðxÞcðxÞ
q 2m
2m
x A Zq
on V ðrÞ .
252
Hajime Tanaka
ðrÞ
Proposition 2.6 (cf. [13, Proposition 2.2]). For each element u of Zq2m , the function eu
ðrÞ
ðrÞ
ðrÞ
is a common eigenfunction of the Al; a ’s, and the eigenvalue of Al; a corresponding to eu
is given by
ðrÞ
ll; a; u ¼
X
eðrÞ
u ðzÞ:
ð2Þ
ðrÞ
z A Ll; a
Proof. Note that
X
ðrÞ
Al; a eðrÞ
u ðxÞ ¼
ðrÞ
xy A Ll; a
eðrÞ
u ðyÞ ¼
X
eðrÞ
u ðx þ zÞ ¼
X
ðrÞ
z A Ll; a
eðrÞ
ðzÞ
euðrÞ ðxÞ:
u
r
ðrÞ
z A Ll; a
ðrÞ
Our next problem is to evaluate the eigenvalues ll; a; u .
Proposition 2.7 (cf. [13, Theorem 2.3]). (i) For 0 c l < r, a A Zp rl and u A Zq2m , we
ðrÞ
ðrlÞ
have ll; a; u ¼ l0; a; u .
(ii) For a A Zq and u A Zq2m , we have
8
<jLðrÞ j;
if u ¼ 0;
0; a
ðrÞ
l0; a; u ¼
: p kð2m1Þ lðrkÞ k ; if u 0 0;
0; a; ð1= p Þu
where k ¼ ordðrÞ
p ðuÞ.
ðrÞ
Proof. (i) This is clear from the definition of Ll; a .
(ii) The assertion is trivial when k ¼ 0 or k ¼ r (i.e. u ¼ 0) therefore, we assume
1 c k < r, so that r d 2.
We write a ¼ a 0 þ p r1 a 00 , with a 0 A Zp r1 and a 00 A Zp . For each z ¼ z 0 þ p r1 z 00 A
2m
00
Zq with z 0 A Zp2m
A Zp2m , we have
r1 and z
0 00
r
fr ðzÞ 1 fr ðz 0 Þ þ p r1 B
r ðz ; z Þ mod p ;
since r d 2. Now, let
ðr1Þ 0
M ¼ fz 0 A Zp2m
ðz Þ ¼ 0; fr ðz 0 Þ 1 a 0 mod p r1 g;
r1 j ordp
and for each z 0 A M let
0 00
00
Nðz 0 Þ ¼ fz 00 A Zp2m j B
r ðz ; z Þ 1 a y mod pg
where fr ðz 0 Þ ¼ a 0 þ p r1 y with y A Zp . Then since z 0 D 0 mod p and B
1 ¼ Br is
0
2m1
non-degenerate, we have jNðz Þj ¼ p
, from which it follows that
Association schemes of a‰ne type over finite rings
ðrÞ
l0; a; u ¼
X
X
253
0
r1 00
eðrÞ
z Þ
u ðz þ p
z 0 A M z 00 A Nðz 0 Þ
1
0
r1 00
¼
e
Br p u; z þ p z
p
z 0 A M z 00 A Nðz 0 Þ
X
X
ðrÞ
ðr1Þ
¼ p 2m1 l0; a; ð1=pÞu :
By repeating the same argument, we obtain the desired result.
r
ðrÞ
By virtue of Proposition 2.7, we only have to evaluate l0; a; u for u A Zq2m with
ordðrÞ
p ðuÞ ¼ 0.
In the case of fields, the character tables of the association schemes of a‰ne type
are described by using the Kloosterman sums ([12], [5]). In the process of studying
ðrÞ
l0; a; u , we will encounter the Kloosterman sums over rings. Let k be a linear character
of the multiplicative group Z
q . Then for a; b A Zq , we define the Kloosterman sum
K ðrÞ ðk j a; bÞ by
K ðrÞ ðk j a; bÞ ¼
X
kðgÞeðrÞ ðag þ bg½1 Þ:
g A Z
q
Note that these sums are completely evaluated by Salié [15] when r > 1 and k ¼ 1
(see also [8]).
We shall need to evaluate the following exponential sum:
X
WðrÞ
g ¼
eðrÞ ð fr ðvÞgÞ
v A Zq2m
for g A Z
q.
r m
ðrÞ
Proposition 2.8. For any element g of Z
q , we have Wg ¼ ð1Þ q .
Proof. First of all, it is easy to see that
X
eðrÞ ðv1 v2 gÞ ¼ q:
v1 ; v2 A Zq
Therefore, if p is odd, then we have to show that
X
eðrÞ ððv12 ev22 ÞgÞ ¼ ð1Þ r q;
v1 ; v2 A Z q
where, as usual, e is a non-square element in Z
p . However, this equality directly fol-
254
Hajime Tanaka
lows from the evaluation of the Gauss sums (cf. [6, p. 26, Theorem 1.5.2], see also
[13, Corollary 2.7]):
8 i pffiffiffi
>
>
>
h;
if h 1 1 mod 4;
<
h1
X
pffiffiffiffiffiffiffi 2
h
gði; hÞ ¼
expð2p 1 ij =hÞ ¼ >
i pffiffiffiffiffiffiffi
>
j¼0
>
h; if h 1 3 mod 4;
:
h
where i and h are any coprime integers with h > 0 and h odd, and
symbol.
If p ¼ 2, then in this case we have to show that
X
oðrÞ
g ¼
s
h
ð3Þ
is the Jacobi
eðrÞ ððv12 þ v1 v2 þ v22 ÞgÞ ¼ ð1Þ r q:
ð4Þ
v1 ; v2 A Zq
Now, let v1 be an odd element of Zq . Then, by Lemma 1.7 (ii) it follows that
fv1 v2 þ v22 j v2 : oddg ¼ fv1 v2 þ v22 j v2 : eveng ¼ 2Zq ;
so that if r > 1, then we have
X
X
eðrÞ ððv12 þ v1 v2 þ v22 ÞgÞ ¼
eðrÞ ððv12 þ v1 v2 þ v22 ÞgÞ ¼ 0:
v2 :even
v2 :odd
In this way, we obtain
oðrÞ
g ¼
X
eðrÞ ð4ðw12 þ w1 w2 þ w22 ÞgÞ ¼ 4oðr2Þ
;
g
w1 ; w2 A Z2 r1
if r d 3. Therefore, (4) follows from an easy calculation:
oð1Þ
g ¼ 2;
oð2Þ
g ¼ 4:
This completes the proof of Proposition 2.8.
r
Theorem 2.9 (cf. [13, Theorem 2.9]). Let a be an element of Zq . Then for any u A Zq2m
with ordðrÞ
p ðuÞ ¼ 0, we have the following.
(i) If r > 1, then we have
ðrÞ
l0; a; u ¼ ð1Þ r p rðm1Þ K ðrÞ ð1 j a; fr ðuÞÞ:
(ii) If r ¼ 1, then we have
Association schemes of a‰ne type over finite rings
255
ð1Þ
l0; a; u ¼ p m1 K ð1Þ ð1 j a; f1 ðuÞÞ d0 ðaÞ
where db ðaÞ is defined by
db ðaÞ ¼
1
0
if a 1 b mod p;
if aDb mod p:
ðrÞ
Proof. We regard l0; a; u as a function in a, then it has the Fourier expansion with respect to the linear characters feðrÞ ðagÞgg A Zq of Zq :
ðrÞ
l0; a; u ¼
1 X ðrÞ
C ðuÞeðrÞ ðagÞ
q gAZ g
q
ðrÞ
for all a A Zq , where the coe‰cients Cg ðuÞ are given by
X
CgðrÞ ðuÞ ¼
X
ðrÞ
l0; b; u eðrÞ ðbgÞ ¼
b A Zq
eðrÞ ðB
r ðu; zÞ þ fr ðzÞgÞ
z A Zq2m
ðrÞ
ordp ðzÞ¼0
We write
ðrÞ
l0; a; u ¼
P
1 P
þ
1
2 ;
q
ð5Þ
where
P
1
¼
X
P
CgðrÞ ðuÞeðrÞ ðagÞ;
First of all, assume r > 1. We evaluate
1
¼
X
CgðrÞ ðuÞeðrÞ ðagÞ:
pag
pjg
P
2
¼
X
P
1.
Setting g ¼ pz with z A Zp r1 , we obtain
X
eðrÞ
u ðzÞ
z A Zq2m
eðr1Þ ðð fr ðzÞ aÞzÞ
z A Zp r1
ðrÞ
ordp ðzÞ¼0
¼ p r1
X
eðrÞ
u ðzÞ
z
r1
summed over z A Zq2m such that ordðrÞ
. If we let
p ðzÞ ¼ 0 and fr ðzÞ 1 a mod p
2m
2m
0
r1 00
00
z ¼ z þ p z with z A Zp r1 and z A Zp , then this sum is equal to
p r1
X
z0
0
eðrÞ
u ðz Þ
X
z 00 A Zp2m
00
eð1Þ ðB
r ðu; z ÞÞ;
256
Hajime Tanaka
where the first sum is over z 0 A Zp r1 such that ordðr1Þ
ðz 0 Þ ¼ 0 and fr1
ðz 0 Þ 1
p
ðrÞ
r1
a mod p . Since ordp ðuÞ ¼ 0, this is in turn equal to 0.
P
½1
Now, we evaluate
uÞg fr ðuÞg½1 if
2 . Since Br ðu; zÞ þ fr ðzÞg ¼ fr ðz þ g
p a g, we have
P
2
¼
X X
pag
e
ðrÞ
ð fr ðvÞgÞ
eðrÞ ðag fr ðuÞg½1 Þ;
v
where the inner sum on the right is over v A Zq2m such that v D g½1 u mod pZq2m .
However, since r > 1, it follows from Remark 2.3 that
X
eðrÞ ð fr ðvÞgÞ ¼ 0;
v
where the sum on the left is over v A Zq2m such that v 1 g½1 u mod pZq2m . Hence, we
have
P
2
¼
X
ðrÞ
½1
WðrÞ
Þ ¼ ð1Þ r q m K ðrÞ ð1 j a; fr ðuÞÞ;
g e ðag fr ðuÞg
pag
by Proposition 2.8.
ðrÞ
By substituting the above evaluations to (5), we conclude that the eigenvalue l0; a; u
is written in the desired form.
ð1Þ
Finally, when r ¼ 1, we can evaluate l0; a; u in exactly the same way, but we omit
the details.
r
To summarize:
Theorem 2.10. For 0 c k < r and b A Zp rk , let
ðrÞ
Vk; b ¼ 0 CeðrÞ
u :
ðrÞ
u A Lk; b
ðrÞ
ðrÞ
Then Vk; b is a maximal common eigenspace of the adjacency operators Al; a ð0 c
l < r, a A Zp rl Þ. Moreover, we have the direct sum decomposition:
ðrÞ
V ðrÞ ¼ V0 l
ðrÞ
0 Vk; b ;
0ck<r
b A Zp rk
ðrÞ
ðrÞ
where V0 ¼ Ce0 is the trivial maximal common eigenspace. With these parameterðrÞ
2m
izations, the ðk; b; l; aÞ-entry pl; a ðk; bÞ of the character table PðGO
2m ðZq Þ; Zq Þ of
ðrÞ
ðrÞ
2m
XðGO
2m ðZq Þ; Zq Þ (i.e. the eigenvalue of Al; a corresponding to Vk; b ) is given by
Association schemes of a‰ne type over finite rings
8
>
p kð2m1Þ ð1Þ rkl pðrklÞðm1Þ K ðrklÞ ð1 j a; bÞ;
>
>
< kð2m1Þ
f p m1 K ð1Þ ð1 j a; bÞ þ d0 ðaÞg;
p
ðrÞ
pl; a ðk; bÞ ¼
>
pðrl1Þð2m1Þ p m1 ðp m þ 1Þ;
>
>
: ðrl1Þð2m1Þ
ðp m1 1Þð p m þ 1Þ;
p
if
if
if
if
257
k þ l < r 1;
k þ l ¼ r 1;
k þ l dr and paa;
k þ l dr and pja:
2m
In particular, XðGO
2m ðZq Þ; Zq Þ is self-dual.
Proof. All but the last statement follow from the above calculations. Obviously,
ðrÞ
2m
x 7! ex ðx A Zq2m Þ defines an isomorphism between XðGO
2m ðZq Þ; Zq Þ and its dual.
r
2m
Example 2.11. The character table P ¼ PðGO
2m ðZ4 Þ; Z4 Þ of XðGO2m ðZ4 Þ;
2m
Z4 Þ ðm > 1Þ is given by
0
1
B1
B
B
B1
B
P¼B
B1
B
B1
B
@1
1
2 3m2 ð2 m þ 1Þ
2 2m1
0
2m1
2
0
2 3m2
2 3m2
2 2m1 ð2 m1 1Þð2 m þ 1Þ 2 3m2 ð2 m þ 1Þ
0
2 2m1
2m1
2
0
2m1
0
2
2 2m1
0
2m1 m1
3m2
2
ð2
1Þ
2
2 2m1 ð2 m1 þ 1Þ
2 3m2
1
2 2m1 ð2 m1 1Þð2 m þ 1Þ 2 m1 ð2 m þ 1Þ ð2 m1 1Þð2 m þ 1Þ
C
2 m1 1
0
2 m1
C
C
2m1
m1
m1
C
2
2
1
2
C
m1
m1
C;
2
1
0
2
C
C
2m1
m1
m1
2
2
1
2
C
C
2m1 m1
m1 m
m1
m
ð2
1Þ
2
ð2 þ 1Þ ð2
1Þð2 þ 1Þ A
2
2 m1 ð2 m þ 1Þ ð2 m1 1Þð2 m þ 1Þ
2 2m1 ð2 m1 þ 1Þ
where the row and column indices are ordered as (0), ð0; 1Þ, ð0; 2Þ, ð0; 3Þ, ð0; 0Þ, ð1; 1Þ,
ð1; 0Þ.
2m
2.3 The character table of X(GOB
2m (Zq ), Zq ). It follows from Proposition 2.4 that
þ
2m
the relations of XðGO2m ðZq Þ; Zq Þ are given as follows.
ðrÞ
ðx; yÞ A R 0 , x ¼ y;
ðrÞ
ðrÞ
ðx; yÞ A Rl; a , x y A Pl; a
ðrÞ
2m
for 0 c l < r and a A Zp rl , where Pl; a is an orbit of GOþ
2m ðZq Þ on Zq defined by
258
Hajime Tanaka
ðrÞ
Pl; a
¼
Zq2m
xA
j ordðrÞ
p ðxÞ
¼ l;
frþ
1
rl
x 1 a mod p
pl
ðrlÞ
¼ f p l u j u A P0; a g:
ðrÞ
ðrÞ
The valencies kl; a ¼ jPl; a j are given by
ðrÞ
kl; a
¼
ðrÞ
jPl; a j
¼
if paa;
pðrl1Þð2m1Þ p m1 ðp m 1Þ;
ðrl1Þð2m1Þ
m1
m
ðp
þ 1Þð p 1Þ; if pja;
p
2m
for 0 c l < r and a A Zp rl . In particular, XðGOþ
2m ðZq Þ; Zq Þ is a symmetric association scheme of class
p r þ p r1 þ þ p ¼
pð p r 1Þ
:
p1
2m
Theorem 2.12. The association scheme XðGOþ
2m ðZq Þ; Zq Þ is self-dual. For k; l A
ðrÞ
f0; 1; . . . ; r 1g, a A Zp rl and b A Zp rk , the ðk; b; l; aÞ-entry pl; a ðk; bÞ of the character
2m
2m
þ
þ
table PðGO2m ðZq Þ; Zq Þ of XðGO2m ðZq Þ; Zq Þ is given by
8
p kð2m1Þ pðrklÞðm1Þ K ðrklÞ ð1 j a; bÞ; if k þ l < r 1;
>
>
>
< kð2m1Þ
f p m1 K ð1Þ ð1 j a; bÞ d0 ðaÞg; if k þ l ¼ r 1;
p
ðrÞ
pl; a ðk; bÞ ¼
>
if k þ l d r and paa;
pðrl1Þð2m1Þ p m1 ð p m 1Þ;
>
>
: ðrl1Þð2m1Þ
ðp m1 þ 1Þð p m 1Þ;
if k þ l d r and pja:
p
2m
2m
þ
Example 2.13. The character table P ¼ PðGOþ
2m ðZ4 Þ; Z4 Þ of XðGO2m ðZ4 Þ; Z4 Þ is
given by
0
1 2 3m2 ð2 m 1Þ 2 2m1 ð2 m1 þ 1Þð2 m 1Þ 2 3m2 ð2 m 1Þ
B1
0
2 2m1
2 2m1
B
B
2m1
B1
0
2
0
B
2m1
2m1
P ¼B
0
2
2
B1
B1
0
2 2m1
0
B
B
3m2
2m1 m1
3m2
@1
2
ð2
þ 1Þ
2
2
1
2 3m2
2 2m1 ð2 m1 1Þ
2 3m2
1
2 2m1 ð2 m1 þ 1Þð2 m 1Þ 2 m1 ð2 m 1Þ ð2 m1 þ 1Þð2 m 1Þ
C
2 m1 1
0
2 m1
C
C
C
2 m1
2 m1 1
2 2m1
C
C;
2 m1 1
0
2 m1
C
C
2m1
m1
m1
2
2
1
2
C
C
2 m1 ð2 m 1Þ ð2 m1 þ 1Þð2 m 1Þ A
2 2m1 ð2 m1 þ 1Þ
2 2m1 ð2 m1 1Þ
2 m1 ð2 m 1Þ
ð2 m1 þ 1Þð2 m 1Þ
Association schemes of a‰ne type over finite rings
259
where the row and column indices are ordered as (0), ð0; 1Þ, ð0; 2Þ, ð0; 3Þ, ð0; 0Þ, ð1; 1Þ,
ð1; 0Þ.
2m
Remark 2.14. We can observe that the character tables of XðGO
2m ðZq Þ; Zq Þ and
2m
þ
XðGO2m ðZq Þ; Zq Þ are closely related with each other. In the case of fields, the char2m
2m
acter table of XðGOþ
2m ðFq Þ; Fq Þ is obtained from that of XðGO2m ðFq Þ; Fq Þ essenm1
m1
tially by the replacement q
7! q
(see [10]). In [4], this phenomenon was
called an Ennola type duality. The situation is slightly complicated, but it seems
possible to regard our examples as a variation of Ennola type duality.
2.4 The character table of X(GO2m+1 (Zq ), Z2m+1
), with q odd. In this subsection,
q
we always assume that q ¼ p r is odd. The relations of the association scheme
XðGO2mþ1 ðZq Þ; Zq2mþ1 Þ are given as follows.
ðrÞ
ðx; yÞ A R 0 , x ¼ y;
ðrÞ
ðrÞ
ðx; yÞ A Rl; a , x y A Xl; a
ðrÞ
for 0 c l < r and a A Zp rl , where Xl; a is an orbit of GO2mþ1 ðZq Þ on Zq2mþ1 defined by
ðrÞ
Xl; a
¼
xA
Zq2mþ1
j ordðrÞ
p ðxÞ
¼ l; fr
1
rl
x 1 a mod p
pl
ðrlÞ
¼ fp l u j u A X0; a g:
ðrÞ
ðrÞ
The valencies kl; a ¼ jXl; a j are given by
ðrÞ
kl; a
8
ðrl1Þ2m
>
p m ðp m þ 1Þ; if paa and a : square;
<p
ðrÞ
ðrl1Þ2m
¼ jXl; a j ¼ p
p m ðp m 1Þ; if paa and a : nonsquare;
>
: ðrl1Þ2m
ð p 2m 1Þ;
if pja
p
ð6Þ
for 0 c l < r and a A Zp rl . In particular, XðGO2mþ1 ðZq Þ; Zq2mþ1 Þ is a symmetric association scheme of class
p r þ p r1 þ þ p ¼
pð p r 1Þ
:
p1
Let w be the quadratic character of Z
p , that is, wðaÞ ¼
have the following:
a
p
for a A Z
p . Then, we
Theorem 2.15. The association scheme XðGO2mþ1 ðZq Þ; Zq2mþ1 Þ is self-dual. For k; l A
ðrÞ
f0; 1; . . . ; r 1g, a A Zp rl and b A Zp rk , the ðk; b; l; aÞ-entry pl; a ðk; bÞ of the character
2mþ1
2mþ1
table PðGO2mþ1 ðZq Þ; Zq
Þ of XðGO2mþ1 ðZq Þ; Zq
Þ is given as follows.
(i) If p 1 1 mod 4, then
260
Hajime Tanaka
ðrÞ
pl; a ðk; bÞ ¼
8 2km ðrklÞð2m1Þ=2 ðrklÞ
p
K
ð1 j a; bÞ;
p
>
>
>
>
>
>
>
>
< p 2km pðrklÞð2m1Þ=2 K ðrklÞ ðw j a; bÞ;
if k þ l < r 1
and r k l : even;
>
>
>
2km
>
>
f pð2m1Þ=2 K ð1Þ ðw j a; bÞ d0 ðaÞg;
>p
>
>
: k ðrÞ ;
l; a
if k þ l ¼ r 1;
if k þ l < r 1
and r k l : odd;
if k þ l d r;
ðrÞ
where kl; a is defined in (6).
(ii) If p 1 3 mod 4, then
8 2km ðrklÞð2m1Þ=2 ðrklÞ
p
K
ð1 j a; bÞ;
p
>
>
>
>
>
>
pffiffiffiffiffiffiffi
>
< p 2km 1pðrklÞð2m1Þ=2 K ðrklÞ ðw j a; bÞ;
if k þ l < r 1 and
r k l : even;
if k þ l < r 1 and
ðrÞ
pl; a ðk; bÞ ¼
r k l : odd;
>
>
pffiffiffiffiffiffiffi ð2m1Þ=2 ð1Þ
>
2km
>
>
p
f
1
p
K
ðw
j
a;
bÞ
þ
d
ðaÞg;
if
k þ l ¼ r 1;
0
>
>
: ðrÞ
kl; a ;
if k þ l d r:
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Zbl 57.0211.01
Received 31 July, 2002
H. Tanaka, Graduate School of Mathematics, Kyushu University, Hakozaki 6-10-1, Fukuoka,
812-8581, Japan
Email: htanaka@math.kyushu-u.ac.jp