Association schemes of a‰ne type over finite rings

(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 ringZq¼Z=qZwhereqis 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 ringZ_q¼Z=qZ, where q¼p^r is a prime power.

Let f :F_qⁿ!F_q be a non-degenerate quadratic form over the finite fieldF_q and OðF_qⁿ;fÞits orthogonal group. Sinceid is contained inOðF_qⁿ;fÞ, the action of the semidirect product OðF_qⁿ;fÞyF_qⁿ on F_qⁿ defines a symmetric association scheme XðOðF_qⁿ;fÞ;F_qⁿÞ. 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 Z_k ¼Z=

kZðkANÞin the same manner. However, by the Chinese remainder theorem, it is enough to consider the case wherekis 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 X_qðn;aÞoverZ_qwithaAZ_q ¼Z_qnpZ_qis defined as the graph with the vertex setZ_qⁿ, and the edge set

E¼ fðx;yÞAZ_qⁿZ_qⁿjdðx;yÞ ¼ag;

* The author is supported in part by a grant from the Japan Society for the Promotion of Science.

wheredðx;yÞAZ_q is the ‘‘distance’’ defined by

dðx;yÞ ¼ ðx1y₁Þ²þ ðx2y₂Þ²þ þ ðxny_nÞ²:

We will show that if q is an odd prime power, then in our language these graphs are part of the relations of XðOðZ_qⁿ;dð;0ÞÞ;Z_qⁿÞ, the association scheme of a‰ne type with respect to the non-degenerate quadratic formdðx;0Þ ¼x₁²þx₂²þ þx_n² (dð;0Þis degenerate ifqis even).

In this paper, we determine the character table of the symmetric association schemeXðOðZ_qⁿ;fÞ;Z_qⁿÞexplicitly for all non-degenerate quadratic forms onZ_qⁿ (for both oddqand evenq). 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 En- nola type dualities observed in [4]. Also, as an immediate consequence of these cal- culations, 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 onZ_qⁿ 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 graphsXqðn;aÞused 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 overZ_q.

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 overZq

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 X¼ ðX;fRig_0cicdÞbe a commutative association scheme with the adjacency matricesA0 ¼I;A1;. . .;Ad, whereIis the identity matrix of degreejxj. The algebra Aof dimensiondþ1, generated byA0;. . .;Ad over the complex number fieldC, is called the Bose–Mesner algebra of X. If we consider the action ofA on the vector spaceV ¼C^jXjindexed by the elements ofX, thenV is decomposed into the direct sum of the maximal common eigenspaces:

V¼V₀?V₁ ? ?V_d;

where V0 is the one-dimensional subspace spanned by the all-one vector. Let Ei:V !Vi be the orthogonal projectionð0cicdÞ. Then the setfE0;E1;. . .;Edg forms another basis ofA, and the base change matrixP¼ ðpiðjÞÞis called thechar- acter tableor thefirst eigenmatrixofX:

Ai¼X^d

j¼0

piðjÞEj ð0cicdÞ

(the ðj;iÞ-entry of Pis piðjÞ). In particular,ki¼ pið0Þis the valency of the regular graph ðX;RiÞ. The second eigenmatrix Q¼ ðqiðjÞÞof X is defined by Q¼ jXjP¹, that is,

jXjEi¼X^d

j¼0

q_iðjÞAj ð0cicdÞ:

The numbers m_i¼q_ið0Þ ¼dimV_i ð0cicdÞare called the multiplicities ofX. No- tice that the first eigenmatrixP, together with the multiplicities ofX, gives complete information of the spectra of the graphsðX;R_iÞ ð1cicdÞ.

Now, assume thatXis symmetric and that the underlying setXhas the structure of an abelian group. We call X a translation association scheme if for 0cicd and zAX we have

ðx;yÞARi) ðxþz;yþzÞARi:

For such an association scheme, there is a natural way to define the dual scheme X¼ ðX;fR_ig_0cicdÞ, where X denotes the character group of X. Namely, we define the relationR_i by

ðm;nÞAR_i ,nm¹ AV_i

(considered as a vector of V). Then, X¼ ðX;fR_ig_0cicdÞ becomes a translation association scheme with the eigenmatricesP¼QandQ¼P(see e.g. [7, §2.10B] or [3, §2.6]). The translation association schemeXis calledself-dualif it is isomorphic to its dualX. In particular, ifXis self-dual, then clearly we haveP¼Q.

1.2 The classification of the non-degenerate quadratic forms overZq.Letq¼p^rwith p prime. Ifa is a unit in Z_q¼Z=qZ, we denote its multiplicative inverse in Z_q by a^½1. Sometimes we identify the ring Z_q with the set f0;1;. . .;q1g, and regard Z_pⁿl ðl<rÞas a subset ofZ_qⁿ.

For a nonzero elementaofZ_q, we denote the largest integerlsuch that p^ldivides aby ord^ðrÞ_p ðaÞ. Conventionally, ord^ðrÞ_p ð0Þis defined to ber. Also, ifx¼ ðx1;x₂;. . .;x_nÞ is an element ofZ_qⁿ, then we define ord^ðrÞ_p ðxÞby

ord^ðrÞ_p ðxÞ ¼ min

1cicnord^ðrÞ_p ðxiÞ:

The reduction of x modulo pZ_qⁿ is denoted by xAZ_pⁿ. If ord^ðrÞ_p ðxÞ>0, then there exists a unique element yofZ_pⁿr1 such thatx¼py, and we write y¼¹_px.

For later use, we prove the following lemma.

Lemma 1.1. Let W be a submodule ofZ_qⁿ.Then W is a direct summand of Z_qⁿ if and only if it is free.

Proof.First, suppose thatWis a direct summand ofZ_qⁿso thatW is projective. Then sinceZ_q is local with the maximal ideal pZ_q,W is free (see e.g. [11, p. 9, Theorem 2.5]).

Conversely, suppose thatW is free. Letff1;f2;. . .;f_kg be a basis ofW. Then it follows that f₁;f₂;. . .;f_k AZ_pⁿ are linearly independent over Zp. In fact, assume thata₁f₁þa₂f₂þ þa_kf_k ¼0 holds for somea₁;a₂;. . .;a_k AZ_q. Then we have

f ¼X

paai

aifiA pZ_qⁿ:

This implies f ¼0, since otherwise p^r1f does not vanish, which is a contradiction.

Thus, we have ai¼0 for all i, as desired. Now, take fkþ1;. . .;fn AZ_qⁿ so that

ff₁;f₂;. . .;f_ngforms a basis ofZ_pⁿ. Then by the well-known Nakayama lemma (see

e.g. [11, p. 8, Theorem 2.2]), f1;f2;. . .;fn span Z_qⁿ. Finally, since jPn

i¼1Z_qfij ¼

jZ_qⁿj ¼qⁿ, f1;f2;. . .;fn must be linearly independent over Z_q. This completes the

proof of Lemma 1.1. r

Corollary 1.2.Let x be an element ofZ_qⁿ.Then Z_qx is a direct summand ofZ_qⁿ if and only iford^ðrÞ_p ðxÞ ¼0.

Let B:Z_qⁿZ_qⁿ!Z_q be a symmetric bilinear form. For a subset U of Z_qⁿ, we define the orthogonal complementU^? ofU by

U^?¼ fxAZ_qⁿjBðx;yÞ ¼0 for all yAUg:

The symmetric bilinear form B is said to benon-degenerateif detðBðei;ejÞÞAZ_q ¼ Z_qnpZq, wherefe1;e2;. . .;eng is the standard basis ofZ_qⁿ, that is,ei is the element of Z_qⁿ with 1 in the i-th component and zero elsewhere. Clearly, this condition is equivalent to saying that the map xAZ_qⁿ7!Bðx;ÞAHomZqðZ_qⁿ;Z_qÞ gives an iso- morphism betweenZ_qⁿand HomZqðZ_qⁿ;ZqÞ.

Aquadratic formonZ_qⁿis a map f :Z_qⁿ!Z_qsatisfying fðaxÞ ¼a²fðxÞ;

fðxþyÞ ¼ fðxÞ þfðyÞ þB_fðx;yÞ;

for any aAZ_q and x;yAZ_qⁿ, where Bf is a symmetric bilinear form on Z_qⁿ. Some- times we call the pair ðZ_qⁿ;fÞa quadratic module over Z_q.

Let ðZ_q^m;f⁰Þ be another quadratic module over Z_q. An isometry s:ðZ_qⁿ;fÞ ! ðZ_q^m;f⁰Þis an injective Zq-linear map such that fðxÞ ¼ f⁰ðsðxÞÞfor allxAZ_qⁿ and sðZ_qⁿÞis a direct summand of Z_q^m. If in addition the isometry s is a linear isomor-

phism, then we say that the two quadratic modules ðZ_qⁿ;fÞ and ðZ_q^m;f⁰Þ are iso- morphic, and writeðZ_qⁿ;fÞGðZ_q^m;f⁰Þ.

The quadratic form f is said to benon-degenerateifB_f is non-degenerate. We de- fine the reductions f :Z_pⁿ!Z_p andB_f :Z_pZ_p!Z_pof f andB_f modulopZ_q, in an obvious manner.

The orthogonal group OðZ_qⁿ;fÞ is the group of all linear transformations on Z_qⁿ that fix f, that is,

OðZ_qⁿ;fÞ ¼ fsAGLðZ_qⁿÞ jfðsðxÞÞ ¼ fðxÞfor allxAZ_qⁿg:

Now we classify all non-degenerate quadratic forms onZ_qⁿ using the classification of those onZ_pⁿ. First of all, we prepare two propositions.

Proposition 1.3(cf. [1, p. 10, Proposition 3.2]).Let f be a quadratic form onZ_qⁿ.If W is a direct summand ofZ_qⁿ such that the restriction fj_W of f to W is non-degenerate, then we haveZ_qⁿ ¼W ?W^?.

Proof. For an element x of Z_qⁿ, define a Z_q-linear map j_x:W!Z_q by j_xðyÞ ¼ Bfðx;yÞ for yAW. Since Bfj_W is non-degenerate, there exists a unique element z of W such that j_xðyÞ ¼Bfðz;yÞ for all yAW, so that we have x¼zþ ðxzÞA WþW^?. Since clearlyWVW^?¼0, we obtain the desired result. r Proposition 1.4 (cf. [1, p. 11, Corollary 3.3]). Let f be a quadratic form on Z_qⁿ. Then,for any orthogonal decomposition Z_pⁿ¼W₁?W₂ with respect to f such that the restriction fj_W1 is non-degenerate, there exists an orthogonal decomposition Z_qⁿ¼W1?W2 such that fj_W1is non-degenerate and WiGWi=pWiði¼1;2Þ.

Proof. Let y1;y2;. . .;yn be a basis of Z_pⁿ such that y1;y2;. . .;yl span W1. For each yi¼ ðyi1;yi2;. . .;yinÞ ð1cicnÞ, take an elementxi¼ ðxi1;xi2;. . .;xinÞofZ_qⁿ such that y_i¼x_i. Since detðyijÞ_1ci;jcn00, we have detðxijÞ_1ci;jcnAZ_q so that x₁;x₂;. . .;x_nform a basis ofZ_qⁿ. Put

W1¼Z_qx1lZ_qx2l lZ_qxl:

Since fj_W1 is non-degenerate, we have detðBfðy_i;y_jÞÞ_1ci;jcl00, from which it fol- lows that detðBfðxi;x_jÞÞ_1ci;jclAZ_q, that is, fj_W1 is non-degenerate. Thus, we have Z_qⁿ¼W1?W2 by Proposition 1.3 whereW2¼W₁^?and clearly,WiGWi=pWiði¼

1;2Þ, as desired. r

It is well-known that the non-degenerate quadratic forms overZ_p are classified as follows:

Theorem 1.5 (cf. [14]). (i)Suppose n¼2m is even.If p is odd, then there are two in- equivalent non-degenerate quadratic forms f₁^þand f₁:

f₁^þðxÞ ¼x₁x₂þ þx2m1x_2m;

f₁ðxÞ ¼x₁x₂þ þx2m3x2m2þx_2m1² ex_2m² for x¼ ðx1;x₂;. . .;x_2mÞAZ_p^2m,whereeis a non-square element ofZ_p.

If p¼2,then there are also two inequivalent non-degenerate quadratic forms f₁^þand f₁:

f₁^þðxÞ ¼x₁x₂þ þx_2m1x_2m;

f₁ðxÞ ¼x1x2þ þx2m3x2m2þx_2m1² þx2m1x2mþx_2m² for x¼ ðx1;x₂;. . .;x_2mÞAZ₂^2m.

(ii) Suppose n¼2mþ1 is odd. If p is odd, then there are two inequivalent non- degenerate quadratic forms f₁ and f₁⁰:

f1ðxÞ ¼x1x2þ þx2m1x2mþx_2mþ1² ; f₁⁰ðxÞ ¼x₁x₂þ þx2m1x_2mþex_2mþ1²

for x¼ ðx1;x2;. . .;x2mþ1ÞAZ_p^2mþ1, where eis a non-square element ofZp, but their orthogonal groups OðZ_p^2mþ1;f₁Þand OðZ_p^2mþ1;f₁⁰Þare isomorphic.

If p¼2,then there is no non-degenerate quadratic form onZ₂^2mþ1.

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 Radf of a qua- dratic form f onZ_pⁿby

Radf ¼ f¹ð0ÞVðZ_pⁿÞ^?;

then using the terminology in [14], f is said to be non-degenerate if Radf ¼0. Our definition agrees with this unless p¼2 andn is odd. If one adopts the definition in [14], then it turns out that there exists exactly one inequivalent non-degenerate qua- dratic form f₁onZ^2mþ1₂ :

f1ðxÞ ¼x1x2þ þx2m1x2mþx²_2mþ1;

forx¼ ðx1;x₂;. . .;x2mþ1ÞAZ₂^2mþ1, which is clearly degenerate in our sense.

For the convenience of the discussion, we call the (free) quadratic module ðZqe1lZ_qe2;fÞof rank two with a distinguished basisfe1;e2gsuch that fðe1Þ ¼a, fðe2Þ ¼b andBfðe1;e2Þ ¼1, the quadratic module of type½a;b. Similarly, we call the quadratic moduleðZqe;fÞof rank one with a distinguished basis feg such that fðeÞ ¼z, the quadratic module of type½z. In order to find the isomorphisms among these quadratic modules, we need the following lemma.

Lemma 1.7.(i)If p is odd,then for any uA pZ_q,there exists a unique element aAZ_q such that a11 modp and a²a1umodq.

(ii)If p¼2,then for any odd t and even u inZ_q,there exist unique even a and odd b inZ_qsuch that a²þta1b²þtb1umodq.

Proof.(i) Letaandbbe elements of 1þpZ_q. Then we havea²a1b²bmodqif and only if ðabÞðaþb1Þ10 modq. Since aþb111 modp, this implies a1bmodq. Thereforea²atakes alluA pZqasaruns through 1þpZq.

(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 anya and b in pZq, the quadratic modules of type½a;b and

½0;0are isomorphic.

(ii)If p¼2,then the quadratic modules of type ½g;dand ½1;1are isomorphic for anyganddinZ_q.

(iii)Suppose p is odd,and letebe a non-square element ofZ_p.Then for eachzAZ_q, the quadratic module of type ½zis isomorphic to that of type [1]or ½e, depending on whetherzis a square or not.

Proof.(i) Letfe1;e₂gbe the distinguished basis of the quadratic module of type½0;0.

By Proposition 1.7 (i) and (ii), there exists uniqueaAZq such thata11 modpand a²a1abmodq. Put

e₁⁰ ¼ae₁þa^½1ae₂ and e₂⁰ ¼be₁þe₂: Then since

det a b

a^½1a 1

¼aa^½1ab11 modp;

fe₁⁰;e₂⁰g therefore forms another basis. Clearly, we have fðe₁⁰Þ ¼a, fðe₂⁰Þ ¼b and Bfðe₁⁰;e₂⁰Þ ¼1.

(ii) Let fe1;e2g be the distinguished basis of the quadratic module of type ½1;1.

Then it follows from Proposition 1.7 (ii) that there exista and unique evenb inZ_q such thata²þa1g1 modqand 3gb²3b1ð2aþ1Þ²d1 modq. Put

e₁⁰ ¼ae1þe2 and e₂⁰ ¼ ð2aþ1Þ^½1ð1ab2bÞe1þbe2: Sincebis even,

det a ð2aþ1Þ^½1ð1ab2bÞ

1 b

!

is odd, so thatfe₁⁰;e₂⁰gis a basis. First, we have fðe₁⁰Þ ¼a²þaþ11gmodq. Also, since

ð2aþ1ÞBfðe₁⁰;e₂⁰Þ ¼2að1ab2bÞ þ ð2aþ1Þabþ ð1ab2bÞ þ2ð2aþ1Þb

¼2aþ1;

we haveB_fðe₁⁰;e₂⁰Þ ¼1. Finally, it follows that

ð2aþ1Þ²fðe₂⁰Þ ¼ ð1ab2bÞ²þ ð2aþ1Þð1ab2bÞbþ ð2aþ1Þ²b²

¼3ða²þaþ1Þb²3bþ1 13gb²3bþ1 modq 1ð2aþ1Þ²dmodq;

so that fðe₂⁰Þ ¼d.

(iii) This is trivial, sinceZ_q is a cyclic group if pis odd. r Combining Theorem 1.5, Proposition 1.4 and Proposition 1.8, we conclude that:

Theorem 1.9. (i)Suppose n¼2m is even.If p is odd, then there are two inequivalent non-degenerate quadratic forms f_r^þand f_ronZ^2m_q :

f_r^þðxÞ ¼x₁x₂þ þx2m1x_2m;

f_rðxÞ ¼x₁x₂þ þx2m3x2m2þx_2m1² ex_2m² for x¼ ðx1;x2;. . .;x2mÞAZ_q^2m,whereeis a non-square element ofZ_p.

If p¼2,then there are also two inequivalent non-degenerate quadratic forms f_r^þand f_ronZ^2m_q :

f_r^þðxÞ ¼x₁x₂þ þx_2m1x_2m;

f_rðxÞ ¼x1x2þ þx2m3x2m2þx_2m1² þx2m1x2mþx_2m² for x¼ ðx1;x2;. . .;x2mÞAZ_q^2m.

(ii) Suppose n¼2mþ1 is odd. If p is odd, then there are two inequivalent non- degenerate quadratic forms fr and f_r⁰ on Z^2mþ1_q :

frðxÞ ¼x1x2þ þx2m1x2mþx_2mþ1² ; f_r⁰ðxÞ ¼x₁x₂þ þx2m1x_2mþex_2mþ1²

for x¼ ðx1;x₂;. . .;x_2mþ1ÞAZ^2mþ1_q ,whereeis a non-square element ofZ_p. If p¼2,then there is no non-degenerate quadratic form onZ_q^2mþ1.

In what follows, we denote the orthogonal groups of f_r^þ, f_rand f_rbyGO^þ_2mðZqÞ, GO_2mðZqÞandGO2mþ1ðZqÞ, respectively. It is easy to see that the orthogonal group OðZ_q^2mþ1;f_r⁰Þis isomorphic toGO2mþ1ðZqÞ.

2 The character tables 2.1 The relations of X(O(Zⁿ_q,f),Z

n

q). As in the introduction, let XðOðZ_qⁿ;fÞ;Z_qⁿÞ denote the symmetric association scheme obtained from the action ofOðZ_qⁿ;fÞyZ_qⁿ onZ_qⁿ. That is, the relations ofXðOðZ_qⁿ;fÞ;Z_qⁿÞare the orbits ofOðZ_qⁿ;fÞyZ_qⁿin its natural action onZ_qⁿZ_qⁿ. Since the stabilizer of an element ofZ_qⁿinOðZ_qⁿ;fÞyZ_qⁿ is isomorphic to OðZ_qⁿ;fÞ, the relations ofXðOðZ_qⁿ;fÞ;Z_qⁿÞare in one-to-one corre- spondence with the orbits ofOðZ_qⁿ;fÞonZ_qⁿ.

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.

Theorem 2.1([9, Satz 5.1]).Let f be a non-degenerate quadratic form onZ_qⁿ.Suppose that W is a direct summand ofZ_qⁿandt:W !Z_qⁿis an isometry.Then there exists an extensionsAOðZ_qⁿ;fÞoft,i.e.sj_W ¼t.

We use Theorem 2.1 to determine the relations ofXðOðZ_qⁿ;fÞ;Z_qⁿÞ.

Lemma 2.2.Let f be a non-degenerate quadratic form onZ_qⁿ.Then for any element x ofZ_qⁿ such thatord^ðrÞ_p ðxÞ ¼0 and aAZ_pl ð0cl<rÞ,there exists an element y ofZ_qⁿ such that x1ymodp^rlZ_qⁿand fðyÞ ¼ fðxÞ þp^rla.

Proof.SinceB_f is non-degenerate andx00, there exists an elementzAZ_qⁿsuch that B_fðx;zÞ00, or equivalently,B_fðx;zÞAZ_q. Notice that for anyb;cAZ_pl we have

bBfðx;zÞ þp^rlb²fðzÞ1cBfðx;zÞ þp^rlc²fðzÞmodp^l if and only if

ðbcÞfBfðx;zÞ þp^rlðbþcÞfðzÞg10 modp^l:

SinceB_fðx;zÞ þp^rlðbþcÞfðzÞD0 modp, this impliesb¼c. Therefore, there exists a unique elementcofZ_p^l such thatcBfðx;zÞ þp^rlc²fðzÞ1amodp^l. Clearly, y¼

xþp^rlczAZ_qⁿ 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 zAZ_qⁿ such that for all vAZ_qⁿ with v1xmodp^rlZ_qⁿ,

fðvþp^rlczÞ ðcAZ_plÞare distinct and

ffðvþp^rlczÞ jcAZ_plg ¼ fðxÞ þ p^rlZ_q:

Proposition 2.4.Let f be a non-degenerate quadratic form onZ_qⁿ.Then for two non- zero elements x and y of Z_qⁿ, there exists an automorphism sAOðZ_qⁿ;fÞ such that sðxÞ ¼ y if and only if ord^ðrÞ_p ðxÞ ¼ord^ðrÞ_p ðyÞ and f_p¹lx

1f_p¹ly

modp^rl where l¼ord^ðrÞ_p ðxÞ.

Proof. The ‘‘only if ’’ part is obvious. Assume l¼ord^ðrÞ_p ðxÞ ¼ord^ðrÞ_p ðyÞ ð<rÞ and f ¹

p^lx

1f ¹

p^ly

modp^rl. Then by Lemma 2.2, there exists an element u of Z_qⁿ such thatu1 ¹

p^lxmodp^rlZ_pⁿl and fðuÞ ¼ f_p¹ly

. SinceZqu andZq 1

p^lyare direct summands of Z_qⁿ by Corollary 1.2, it follows from Theorem 2.1 that there exists sAOðZ_qⁿ;fÞsuch thatsðuÞ ¼ _p¹ly. Then we havesðxÞ ¼ p^lsðuÞ ¼ y. r In the next subsection, we calculate the character table ofXðGO_2mðZqÞ;Z_q^2mÞ. The discussions are almost parallel to those in Medrano et al. [13] and DeDeo [8]. In §2.3 and §2.4, the results for XðGO^þ_2mðZqÞ;Z^2m_q Þ and XðGO2mþ1ðZqÞ;Z_q^2mþ1Þ are stated without detailed proofs.

2.2 The character table of X(GO^C_2m(Zq),Z^2m_q ). By Proposition 2.4, the relations of XðGO_2mðZqÞ;Z^2m_q Þare given as follows.

ðx;yÞAR^ðrÞ₀ ,x¼y;

ðx;yÞAR^ðrÞ_l;a,xyAL^ðrÞ_l;a

for 0cl<randaAZ_p^rl, whereL^ðrÞ_l;a is an orbit ofGO_2mðZqÞonZ_q^2m defined by L^ðrÞ_l;a¼ xAZ_q^2mjord^ðrÞ_p ðxÞ ¼l;f_r 1

p^lx

1amodp^rl

¼ fp^lujuAL^ðrlÞ_0;a g:

Notice that thek^ðrÞ_l;a¼ jL^ðrÞ_l;ajare the valencies ofXðGO_2mðZqÞ;Z_q^2mÞ.

Proposition 2.5.For0cl<r and aAZ_prl,we have

k^ðrÞ_l;a¼ jL^ðrÞ_l;aj ¼ p^{ðrl1Þð2m1Þ}p^m1ðp^mþ1Þ; if paa;

p^{ðrl1Þð2m1Þ} ðp^m11Þðp^mþ1Þ; if pja:

In particular,XðGO_2mðZqÞ;Z^2m_q Þis a symmetric association scheme of class p^rþp^r1þ þp¼pðp^r1Þ

p1 if m>1,and of class

ðp^rp^r1Þ þ ðp^r1p^r2Þ þ þ ðp1Þ ¼ p^r1 if m¼1.

Proof.SincejL^ðrÞ_l;aj ¼ jL^ðrlÞ_0;a j, we only have to prove the equality whenl¼0. Ifr¼1, then it is easy to see that the assertion is true (cf. [10, §3.3]). Now, suppose r>1.

Then for an elementxofZ_q^2m, we have ord^ðrÞ_p ðxÞ ¼0 if and only ifx00, from which it follows that

jfxAZ^2m_q jord^ðrÞ_p ðxÞ ¼0;f_rðxÞ1amodpgj

¼ p^ðr1Þ2mp^m1ðp^mþ1Þ; if paa;

p^ðr1Þ2m ðp^m11Þðp^mþ1Þ; if pja.

ð1Þ

By Remark 2.3,jL^ðrÞ_0;ajis obtained by dividing the right-hand side of (1) byp^r1. r Medrano et al. [13] and DeDeo [8] determined the graph spectra of the graphs ðR^ðrÞ_0;a;Z_q^2mÞ ðaAZ_qÞcompletely for many cases. We apply this method in the calcu- lation of the character table ofXðGO_2mðZqÞ;Z^2m_q Þ.

Let V^ðrÞ be the complex vector space of the functions j:Z_q^2m!C. For each 0cl<randaAZ_prl, we define theadjacency operator A^ðrÞ_l;aonV^ðrÞby

A^ðrÞ_l;ajðxÞ ¼ X

xyAL^ðrÞ_l;a

jðyÞ

for alljAV^ðrÞ. We will decomposeV^ðrÞinto the direct sum of the maximal common eigenspaces of theA^ðrÞ_l;a’s.

For brevity, we denote the associated bilinear formB_frof f_r simply byB_r. Also, we use the notation

e^ðrÞðaÞ ¼expð2p ffiffiffiffiffiffiffi p1

a=p^rÞ

foraAZ_q. For eachuAZ_q^2m, define a linear charactere^ðrÞu :Z_q^2m!Cby e^ðrÞ_u ðxÞ ¼e^ðrÞðB_rðu;xÞÞ

forxAZ_q^2m. Then, sinceB_r is non-degenerate, thee^ðrÞu ’s are distinct and they form an orthonormal basis ofV^ðrÞwith respect to the inner product

ðj;cÞ ¼ 1 q^2m

X

xAZ_q^2m

jðxÞcðxÞ

onV^ðrÞ.

Proposition 2.6(cf. [13, Proposition 2.2]).For each element u ofZ_q^2m,the function e^ðrÞ_u is a common eigenfunction of the A^ðrÞ_l;a’s,and the eigenvalue of A^ðrÞ_l;acorresponding to e^ðrÞu

is given by

l^ðrÞ_l;a;u¼ X

zAL^ðrÞ_l;a

e^ðrÞ_u ðzÞ: ð2Þ

Proof.Note that

A^ðrÞ_l;ae^ðrÞ_u ðxÞ ¼ X

xyAL^ðrÞ_l;a

e^ðrÞ_u ðyÞ ¼ X

zAL^ðrÞ_l;a

e^ðrÞ_u ðxþzÞ ¼ X

zAL^ðrÞ_l;a

e^ðrÞ_u ðzÞ

e^ðrÞ_u ðxÞ: r

Our next problem is to evaluate the eigenvaluesl^ðrÞ_l;a;u.

Proposition 2.7 (cf. [13, Theorem 2.3]). (i) For 0cl<r, aAZ_prl and uAZ^2m_q , we havel^ðrÞ_l;a;u ¼l^ðrlÞ_0;a;u.

(ii)For aAZ_qand uAZ_q^2m,we have

l^ðrÞ_0;a;u¼ jL^ðrÞ_0;aj; if u¼0;

p^kð2m1Þl^ðrkÞ_0;a;ð1=pkÞu; if u00;

8<

:

where k¼ord^ðrÞ_p ðuÞ.

Proof. (i) This is clear from the definition ofL^ðrÞ_l;a.

(ii) The assertion is trivial whenk¼0 or k¼r(i.e. u¼0) therefore, we assume 1ck<r, so thatrd2.

We writea¼a⁰þp^r1a⁰⁰, witha⁰AZ_pr1 anda⁰⁰AZ_p. For eachz¼z⁰þp^r1z⁰⁰A Z_q^2mwithz⁰AZ_p^2mr1 andz⁰⁰AZ^2m_p , we have

f_rðzÞ1f_rðz⁰Þ þp^r1B_rðz⁰;z⁰⁰Þmodp^r; sincerd2. Now, let

M¼ fz⁰AZ_p^2mr1jord^ðr1Þ_p ðz⁰Þ ¼0;f_rðz⁰Þ1a⁰modp^r1g;

and for eachz⁰AMlet

Nðz⁰Þ ¼ fz⁰⁰AZ^2m_p jB_rðz⁰;z⁰⁰Þ1a⁰⁰ymodpg

where f_rðz⁰Þ ¼a⁰þp^r1y with yAZ_p. Then since z⁰D0 modp and B₁ ¼B_r is non-degenerate, we havejNðz⁰Þj ¼ p^2m1, from which it follows that

l^ðrÞ_0;a;u¼ X

z⁰AM

X

z⁰⁰ANðz⁰Þ

e^ðrÞ_u ðz⁰þp^r1z⁰⁰Þ

¼ X

z⁰AM

X

z⁰⁰ANðz⁰Þ

e^ðrÞ B_r p1

pu;z⁰þp^r1z⁰⁰

¼ p^2m1l^ðr1Þ_0;a;ð1=pÞu:

By repeating the same argument, we obtain the desired result. r By virtue of Proposition 2.7, we only have to evaluate l^ðrÞ_0;a;u for uAZ_q^2m with ord^ðrÞ_p ðuÞ ¼0.

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 l^ðrÞ_0;a;u, we will encounter the Kloosterman sums over rings. Letkbe a linear character of the multiplicative group Z_q. Then for a;bAZ_q, we define the Kloosterman sum K^ðrÞðkja;bÞby

K^ðrÞðkja;bÞ ¼ X

gAZ_q

kðgÞe^ðrÞðagþbg^½1Þ:

Note that these sums are completely evaluated by Salie´ [15] when r>1 and k¼1 (see also [8]).

We shall need to evaluate the following exponential sum:

W^ðrÞ_g ¼ X

vAZ_q^2m

e^ðrÞðf_rðvÞgÞ

forgAZ_q.

Proposition 2.8.For any elementgofZ_q,we haveW^ðrÞ_g ¼ ð1Þ^rq^m. Proof.First of all, it is easy to see that

X

v1;v2AZq

e^ðrÞðv1v2gÞ ¼q:

Therefore, if pis odd, then we have to show that X

v1;v2AZq

e^ðrÞððv₁²ev₂²ÞgÞ ¼ ð1Þ^rq;

where, as usual,eis a non-square element inZ_p. However, this equality directly fol-

lows from the evaluation of the Gauss sums (cf. [6, p. 26, Theorem 1.5.2], see also [13, Corollary 2.7]):

gði;hÞ ¼X^h1

j¼0

expð2p ffiffiffiffiffiffiffi p1

ij²=hÞ ¼ i h ffiffiffi

ph

; if h11 mod 4;

i h ffiffiffiffiffiffiffi

ph

; if h13 mod 4;

8>

>>

<

>>

>:

ð3Þ

where iandhare any coprime integers withh>0 andhodd, and _h^s

is the Jacobi symbol.

If p¼2, then in this case we have to show that o^ðrÞ_g ¼ X

v1;v2AZq

e^ðrÞððv²₁þv₁v₂þv₂²ÞgÞ ¼ ð1Þ^rq: ð4Þ

Now, letv1 be an odd element ofZ_q. Then, by Lemma 1.7 (ii) it follows that fv1v2þv₂²jv2:oddg ¼ fv1v2þv₂²jv2 :eveng ¼2Zq;

so that ifr>1, then we have X

v2:odd

e^ðrÞððv₁²þv1v2þv₂²ÞgÞ ¼ X

v2:even

e^ðrÞððv²₁þv1v2þv₂²ÞgÞ ¼0:

In this way, we obtain

o^ðrÞ_g ¼ X

w1;w2AZ₂r1

e^ðrÞð4ðw₁²þw₁w₂þw₂²ÞgÞ ¼4o^ðr2Þ_g ;

ifrd3. Therefore, (4) follows from an easy calculation:

o^ð1Þ_g ¼ 2; o^ð2Þ_g ¼4:

This completes the proof of Proposition 2.8. r

Theorem 2.9(cf. [13, Theorem 2.9]).Let a be an element ofZ_q.Then for any uAZ_q^2m withord^ðrÞ_p ðuÞ ¼0,we have the following.

(i)If r>1,then we have

l^ðrÞ_0;a;u¼ ð1Þ^rp^rðm1ÞK^ðrÞð1ja;f_rðuÞÞ: (ii)If r¼1,then we have

l^ð1Þ_0;a;u¼ p^m1K^ð1Þð1ja;f₁ðuÞÞ d₀ðaÞ whered_bðaÞis defined by

dbðaÞ ¼ 1 if a1bmodp;

0 if aDbmodp:

Proof.We regardl^ðrÞ_0;a;u as a function ina, then it has the Fourier expansion with re- spect to the linear charactersfe^ðrÞðagÞg_gAZqofZq:

l^ðrÞ_0;a;u¼1 q

X

gAZq

C_g^ðrÞðuÞe^ðrÞðagÞ

for allaAZ_q, where the coe‰cientsCg^ðrÞðuÞare given by C_g^ðrÞðuÞ ¼ X

bAZq

l^ðrÞ_0;b;ue^ðrÞðbgÞ ¼ X

zAZ^2m_q ord^ðrÞpðzÞ¼0

e^ðrÞðB_rðu;zÞ þf_rðzÞgÞ

We write

l^ðrÞ_0;a;u¼1 q

P

1þP

2

; ð5Þ

where

P

1¼X

pjg

C_g^ðrÞðuÞe^ðrÞðagÞ; P

2¼X

pag

C_g^ðrÞðuÞe^ðrÞðagÞ:

First of all, assumer>1. We evaluateP

1. Settingg¼pzwithzAZ_pr1, we obtain P

1¼ X

zAZ_q^2m ord^ðrÞp ðzÞ¼0

e^ðrÞ_u ðzÞ X

zAZ_{p r}1

e^ðr1Þððf_rðzÞ aÞzÞ

¼ p^r1X

z

e^ðrÞ_u ðzÞ

summed over zAZ_q^2m such that ord^ðrÞ_p ðzÞ ¼0 and f_rðzÞ1amodp^r1. If we let z¼z⁰þp^r1z⁰⁰withzAZ_p^2mr1 andz⁰⁰AZ_p^2m, then this sum is equal to

p^r1X

z⁰

e^ðrÞ_u ðz⁰Þ X

z⁰⁰AZ_p^2m

e^ð1ÞðB_rðu;z⁰⁰ÞÞ;

where the first sum is over z⁰AZ_pr1 such that ord^ðr1Þ_p ðz⁰Þ ¼0 and f_r1 ðz⁰Þ1 amodp^r1. Since ord^ðrÞ_p ðuÞ ¼0, this is in turn equal to 0.

Now, we evaluate P

2. Since B_rðu;zÞ þf_rðzÞg¼ f_rðzþg^½1uÞgf_rðuÞg^½1 if pag, we have

P

2¼X

pag

X

v

e^ðrÞðf_rðvÞgÞ

e^ðrÞðagf_rðuÞg^½1Þ;

where the inner sum on the right is over vAZ_q^2m such that vDg^½1umodpZ_q^2m. However, sincer>1, it follows from Remark 2.3 that

X

v

e^ðrÞðf_rðvÞgÞ ¼0;

where the sum on the left is overvAZ_q^2m such thatv1g^½1umodpZ_q^2m. Hence, we have

P

2 ¼X

pag

W^ðrÞ_g e^ðrÞðagf_rðuÞg^½1Þ ¼ ð1Þ^rq^mK^ðrÞð1ja;f_rðuÞÞ;

by Proposition 2.8.

By substituting the above evaluations to (5), we conclude that the eigenvaluel^ðrÞ_0;a;u is written in the desired form.

Finally, whenr¼1, we can evaluatel^ð1Þ_0;a;u in exactly the same way, but we omit

the details. r

To summarize:

Theorem 2.10.For0ck<r and bAZ_p^rk,let

V_k;b^ðrÞ¼ 0

uAL^ðrÞ_k;b

Ce^ðrÞ_u :

Then V_k;^ðrÞ_b is a maximal common eigenspace of the adjacency operators A^ðrÞ_l;að0c l<r,aAZ_p^rlÞ.Moreover,we have the direct sum decomposition:

V^ðrÞ¼V₀^ðrÞl 0

0ck<r bAZ_{p r}k

V_k;b^ðrÞ;

where V₀^ðrÞ¼Ce^ðrÞ₀ is the trivial maximal common eigenspace. With these parameter- izations, the ðk;b;l;aÞ-entry p^ðrÞ_l;aðk;bÞ of the character table PðGO_2mðZqÞ;Z_q^2mÞ of XðGO_2mðZqÞ;Z_q^2mÞ(i.e. the eigenvalue of A^ðrÞ_l;a corresponding to V_k;b^ðrÞ)is given by

p^ðrÞ_l;aðk;bÞ ¼

p^kð2m1Þ ð1Þ^rklp^{ðrklÞðm1Þ}K^ðrklÞð1ja;bÞ; if kþl<r1;

p^kð2m1Þ fp^m1K^ð1Þð1ja;bÞ þd0ðaÞg; if kþl¼r1;

p^{ðrl1Þð2m1Þ}p^m1ðp^mþ1Þ; if kþldr and paa;

p^{ðrl1Þð2m1Þ} ðp^m11Þðp^mþ1Þ; if kþldr and pja:

8>

>>

<

>>

>:

In particular,XðGO_2mðZqÞ;Z^2m_q Þis self-dual.

Proof. All but the last statement follow from the above calculations. Obviously, x7!e^ðrÞ_x ðxAZ^2m_q Þdefines an isomorphism betweenXðGO_2mðZqÞ;Z^2m_q Þand its dual.

r Example 2.11. The character table P¼PðGO_2mðZ4Þ;Z₄^2mÞ of XðGO_2mðZ4Þ;

Z₄^2mÞ ðm>1Þis given by

P¼ 0 BB BB BB BB BB

@

1 2^3m2ð2^mþ1Þ 2^2m1ð2^m11Þð2^mþ1Þ 2^3m2ð2^mþ1Þ

1 2^2m1 0 2^2m1

1 0 2^2m1 0

1 2^2m1 0 2^2m1

1 0 2^2m1 0

1 2^3m2 2^2m1ð2^m11Þ 2^3m2

1 2^3m2 2^2m1ð2^m1þ1Þ 2^3m2

2^2m1ð2^m11Þð2^mþ1Þ 2^m1ð2^mþ1Þ ð2^m11Þð2^mþ1Þ

0 2^m1 2^m11

2^2m1 2^m1 2^m11

0 2^m1 2^m11

2^2m1 2^m1 2^m11

2^2m1ð2^m11Þ 2^m1ð2^mþ1Þ ð2^m11Þð2^mþ1Þ 2^2m1ð2^m1þ1Þ 2^m1ð2^mþ1Þ ð2^m11Þð2^mþ1Þ

1 CC CC CC CC CC A

;

where the row and column indices are ordered as (0),ð0;1Þ,ð0;2Þ,ð0;3Þ,ð0;0Þ,ð1;1Þ, ð1;0Þ.

2.3 The character table ofX(GO^B_2m(Zq),Z^2m_q ).It follows from Proposition 2.4 that the relations ofXðGO_2m^þ ðZqÞ;Z_q^2mÞare given as follows.

ðx;yÞAR^ðrÞ₀ ,x¼y;

ðx;yÞAR^ðrÞ_l;a,xyAP^ðrÞ_l;a

for 0cl<randaAZ_prl, whereP^ðrÞ_l;a is an orbit ofGO^þ_2mðZqÞonZ_q^2mdefined by