TatsuyaFujisaki q ) Afour-classassociationschemederivedfromahyperbolicquadricinPG(3,

A four-class association scheme derived from a hyperbolic quadric in PG(3, q)

Tatsuya Fujisaki

(Communicated by E. Bannai)

Abstract.We prove the existence of a four-class association scheme on the set of external lines with respect to a hyperbolic quadric of PGð3;qÞwhereqd4 is a power of 2. This result is an analogue of the one by Ebert, Egner, Hollmann and Xiang. Taking a quotient of this associa- tion scheme yields a strongly regular graph of Latin square type. We show that this strongly regular graph can also be obtained by a generalization of the construction given by Mathon.

1 Introduction

In the paper [5], Ebert, Egner, Hollmann and Xiang constructed a four-class sym- metric association scheme by using the set of secant lines with respect to an ovoidO of PGð3;qÞforqd4 a power of 2. We can regard this association scheme as defined on the set of external lines by taking the null polarity with respect toO. In this paper, we consider an analogous construction by a hyperbolic quadric. We construct a four- class symmetric association scheme by using the set of external lines with respect to a hyperbolic quadric of PGð3;qÞ. Each relation is invariant under the action of the orthogonal groupO^þð4;qÞbut the set of relations is not the set of orbitals on the set of external lines. Indeed, there are more orbitals than relations. Moreover, a quo- tient of this association scheme forms a strongly regular graph of Latin square type.

We also prove that this strongly regular graph is isomorphic to the one constructed from a direct product of a pseudo-cyclic symmetric association scheme defined by the action of SLð2;qÞon the right cosets SLð2;qÞ=Oð2;qÞ, which is a generalization of the construction given by Mathon [10]. This isomorphism is obtained by an iso- morphism between SLð2;qÞ²andW^þð4;qÞ.

2 Association schemes, strongly regular graphs and projective spaces LetX be a finite set and letfRig_0cicd be relations onX, that is, subsets ofXX.

ThenX¼ ðX;fRig_0cicdÞis called a d-class symmetric association schemeif the fol- lowing conditions are satisfied.

1. fRig_0cicd is a partition ofXX.

2. R₀is diagonal, that is,R₀¼ fðx;xÞ jxAXg.

3. fðy;xÞ j ðx;yÞAR_ig ¼R_i for anyi.

4. For anyi;j;kAf0;1;. . .;dg, p_ij^k:¼ jfzAXj ðx;zÞAR_i;ðy;zÞAR_jgjis independent of the choice ofðx;yÞinR_k.

ForiAf0;. . .;dg, letAi be the adjacency matrix of the relation Ri, that is,Ai is

indexed byX and

ðAiÞ_xy:¼ 1 if ðx;yÞAR_i; 0 if ðx;yÞBR_i:

Then we have

AiAj¼X^d

k¼0

p_ij^kAk

for anyi;jAf0;. . .;dg. SoA0;A1;. . .;Ad form a basis of the commutative algebra generated byA0;A1;. . .;Ad over the complex field (which is called the Bose–Mesner algebra of X). Moreover this algebra has a unique basisE0;E1;. . .;Ed of primitive idempotents. One of the primitive idempotents isjXj¹J whereJis the matrix whose entries are all 1. So we may assume E0¼ jXj¹J. Let P¼ ðpjðiÞÞ_0ci;jcd be the ma- trix defined by

ðA0 A1. . .A_dÞ ¼ ðE0 E1. . .E_dÞP:

We callPthe first eigenmatrix ofX. Note thatfpjðiÞ j0cicdgis the set of eigen- values ofAj. The first eigenmatrix satisfies the orthogonality relation:

X^d

n¼0

1

k_npnðiÞpnðjÞ ¼jXj m_i dij;

where k_i¼p⁰_ii and m_i¼rankE_i. We say that X is pseudo-cyclic if there exists an integer msuch that rankE_i¼mfor alliAf1;. . .;dg. Note that in this case, jXj ¼ dmþ1 andk_i¼p_ii⁰¼mfor alliAf1;. . .;dg(see [1, p. 76]).

LetGbe a finite group andKbe a subgroup ofG. ThenGacts naturally on the set G=KG=Kwith orbitalsR0;R1;. . .;Rd, where we letR0¼ fðgK;gKÞ jgKAG=Kg.

If all orbitals are self-paired, thenX¼ ðG=K;fRig_0cicdÞforms a symmetric associa- tion scheme. We denote this association scheme byXðG;KÞ.

For a strongly regular graph with parametersðn;k;l;mÞ, one of the eigenvalues of its adjacency matrix isk, and the others y1;y2 are the solutions ofx²þ ðmlÞxþ ðmkÞ ¼0. We can identify the pair of a strongly regular graph and its complement with a two-class symmetric association scheme whose first eigenmatrix is

2 64

1 k nk1

1 y1 1y1

1 y2 1y2

3

75 ð1Þ

In the paper [10], Mathon constructed a strongly regular graph from the pseudo- cyclic symmetric association schemeXðSLð2;8Þ;Oð2;8ÞÞ. The next lemma is a gen- eralization of this construction, due to Brouwer and Mathon [2]. Godsil [7] remarks that it can also be proved by Koppinen’s identity [9] (see also [6, Theorem 2.4.1]).

Lemma 2.1.LetX¼ ðX;fRig_0cicdÞbe a pseudo-cyclic symmetric association scheme on dmþ1points.Then the graph DðXÞwhose vertex set is XX,where two distinct verticesðx;yÞandðx⁰;y⁰Þare adjacent if and only ifðx;x⁰Þ;ðy;y⁰ÞARifor some i00, is a strongly regular graph of Latin square type with parameters

ðjXj²;mðjXj 1Þ;jXj þmðm3Þ;mðm1ÞÞ:

Proof.The direct product ofXisðXX;fRijg_0ci;jcdÞ, where Rij:¼ fððx;yÞ;ðx⁰;y⁰ÞÞ j ðx;x⁰ÞARi;ðy;y⁰ÞARjg:

If P is the first eigenmatrix of X, then PnP is the first eigenmatrix of ðXX, fRijg_0ci;jcdÞ. The edge set ofDðXÞis defined to be6^d

j¼1Rjj. Then the eigenvalues of the adjacency matrix ofDðXÞare

X^d

j¼1

p_jðiÞp_jði⁰Þ j0ci;i⁰cd

:

Since X is psuedo-cyclic, k₀¼m₀¼1, k_j¼m_i¼m for i;j00. Hence the ortho- gonality relation implies

X^d

j¼1

pjðiÞpjði⁰Þ ¼mjXj mi

dii⁰m¼

mðjXj 1Þ if i¼i⁰¼0;

jXj m if i¼i⁰00;

m if i0i⁰: 8<

:

ThereforeDðXÞhas three eigenvalues. This implies thatDðXÞis strongly regular. The

parameters ofDðXÞcan easily be calculated. r

In Lemma 2.1, if X¼XðG;KÞfor some finite group G and its subgroupK, then DðXÞhas the following geometric interpretation.

Lemma 2.2. Suppose that a finite group G and its subgroup K form a pseudo-cyclic symmetric association scheme X¼XðG;KÞ. Then the graph DðXÞ of Lemma 2.1 is isomorphic to the collinearity graph of the coset geometryðG²=K²;G²=DðGÞ;Þwhere DðGÞ:¼ fðx;xÞ jxAGgand for x1;x2;y1;y2AG,ðx1;x2ÞK² ðy1;y2ÞDðGÞif and only ifðx1;x₂ÞK²Vðy₁;y₂ÞDðGÞ ¼q.

Proof.Since each relation ofXðG;KÞis an orbital of the action ofGonG=KG=K, two pairsðx1K;y₁KÞ,ðx2K;y₂KÞare adjacent in the graphDðXðG;KÞÞif and only if there existswAGsuch that y₁K¼wx₁K,y₂K¼wx₂K. On the other hand, two pairs ðx1;y1ÞK²,ðx2;y2ÞK² are adjacent in the collinearity graph ofðG²=K²;G²=DðGÞ;Þ if and only ifðx¹₁ x2;y¹₁ y2Þis inK²DðGÞK² (cf. [4, p. 15]).

Forx1;x2;y1;y2AG,

ðx¹₁ x₂;y¹₁ y₂ÞAK²DðGÞK² ,x¹₁ x₂;y¹₁ y₂AKwK for somewAG;

,x¹₁ x2AKy¹₁ y2K

,y1kx¹₁ ¼y2k⁰x¹₂ for somek;k⁰AK;

,y1Awx1K;y2Awx2Kfor somewAG;

Hence the mappingG=KG=KCðxK;yKÞ 7! ðx;yÞK²AG²=K²is an isomorphism

between the above two graphs. r

For the rest of this section, we recall some terminology on finite projective spaces.

In this paper, letqbe a power of 2 and let PGð3;qÞbe the three-dimensional projec- tive space over GFðqÞ. For a non-degenerate quadratic form Qon GFðqÞ⁴, we say that a point p¼hviissingularifQðvÞ ¼0, and we call the set of singular points a quadric. For a set of pointsX, we say that a linelisexternal(respectivelysecant) to the setX if the number of points inlVX is 0 (respectively 2).

It is well known that there are two types of non-degenerate quadratic forms on GFðqÞ⁴, which are called elliptic type or hyperbolic type. For a pointp, denote byp^? the orthogonal complement of pwith respect to the symmetric bilinear form obtained fromQ. Define for a linelor a planep,l^?:¼7

pAlp^?,p^?:¼7

pApp^?.

For a hyperbolic quadric in PGð3;qÞ, since qis even, the polarity?is a null po- larity, that is, if p is a point, then pAp^?. More precisely, if p is on the hyperbolic quadric, then p^?is the plane determined by the two generators of the quadric through p. If p is not on the quadric, then through p there areqþ1 tangent lines to the quadric and theseqþ1 lines are coplanar. The plane determined by theseqþ1 tan- gent lines is p^?. For a line l, we have l^?¼ fp^?jlJpg. If l is external, then since every planepcontainingl satisfies the pointp^?is nonsingular and not onl, the line l^?is also external to the quadric and skew tol. On the other hand, for an external line lto an ovoid, the linel^?is skew toland secant to the ovoid (see [8, pp. 24–26]).

A canonical form of the quadratic form of hyperbolic type is Qðx1;x₂;x₃;x₄Þ ¼x₁x₄þx₂x₃:

Denote byW^þð4;qÞthe commutator group of the orthogonal group defined from the aboveQ.

3 Main results

A four-class symmetric association scheme on the set of secant lines with respect to any ovoid was constructed:

Theorem 3.1 ([5]). Let q¼2^fd4.Then the following relations on the set of secant lines ofPGð3;qÞwith respect to an ovoid

R1¼ fðl;mÞ jlVm is a singular pointg R2¼ fðl;mÞ jlVm is a nonsingular pointg R₃¼ fðl;mÞ jl^?Vm0 qg

R4¼ fðl;mÞ jlVm¼q;l^?Vm¼qg

and the diagonal relation R₀ define a four-class symmetric association scheme.

We can regard the above association scheme as defined on the set of external lines.

The relations R₁;R₂;R₃ andR₄ correspond to the following relations on the set of external lines

fðl;mÞ jhl;mi^? is a singular pointg; fðl;mÞ jhl;mi^? is a nonsingular pointg; fðl;mÞ jl^?Vm0 qg;

fðl;mÞ jlVm¼q;l^?Vm¼qg;

respectively.

In the paper [5], a plane pis called tangent if its orthogonal complement is a sin- gular point.

For a hyperbolic quadric, we can construct a four-class symmetric association scheme similar to the above one. Let Lbe the set of external lines with respect to a hyperbolic quadric in PGð3;qÞ.

Theorem 3.2. Let q¼2^f d4. Then the following relations on the set L of external lines ofPGð3;qÞwith respect to a hyperbolic quadric

R1¼ fðl;mÞ jlVm is a pointg R₂¼ fðl;mÞ jm¼l^?g

R₃¼ fðl;mÞ jl^?Vm is a pointg R4¼ fðl;mÞ jlVm¼q;l^?Vm¼qg

and the diagonal relation R₀ define a four-class symmetric association scheme.

Moreover we can construct a strongly regular graph from this symmetric association scheme by taking a quotient.

Theorem 3.3. Let G¼G_q ðq¼2^fd4Þ be the graph with vertex set ffl;l^?g jlALg, where two distinct vertices ofG,fl;l^?g,fm;m^?gare adjacent if and only if lVm0 q or lVm^?0 q.Then Gis a strongly regular graph of Latin square type with param- eters

v¼1

4q²ðq1Þ²; k¼1

2ðq2Þðqþ1Þ²; l¼1

2ð3q²3q4Þ; m¼qðqþ1Þ:

Note thatlVm0 qis equivalent tol^?Vm^?0 q, andlVm^?0 qis equivalent to l^?Vm0 q. So the adjacency inGis well-defined.

4 Proof of Theorem 3.2

To prove Theorem 3.2, we recall some facts about PGð3;qÞwith a hyperbolic quadric from Hirschfeld’s book [8, §15–III]. From now on, putq¼2^f d4. LetPbe the set of planes whose orthogonal complement is a nonsingular point.

Proposition 4.1.For a hyperbolic quadric inPGð3;qÞ,the following statements hold.

(i) A plane containing an external line is inP.

(ii) The number of external lines is q²ðq1Þ²=2 and there are qþ1 planes of P containing a given external line.

(iii) The number of planes inPis qðq²1Þand there are qðq1Þ=2external lines in a given plane ofP.

(iv) ForpAP,there is no external line throughp^? onp.For a nonsingular point p of pdistinct fromp^?,there are q=2external lines through p onp.

(v) There are qðq1Þ=2external lines through a given nonsingular point.

(Remark: whenqis an odd prime power, (i), (ii), (iii) and (v) also hold. For a plane pofP,p^?is not inp.)

First we show that the relationsR0;. . .;R4 form a partition of LL. It is clear that any pairðl;mÞofLLis in one offRig_0cic4. Since any external linelis skew tol^?;R1 and R2 have no intersection. Suppose that l;mALsatisfy that l meetsm.

Then the pointhl;mi^? is onl^?, somis skew tol^?by Proposition 4.1 (iv). HenceR1

andR3 have no intersection. ThereforefRig_0cic4 is a partition ofLL.

Next we show that each relation is symmetric. It is clear that R₁;R₂ andR₄ are symmetric. Ifðl;mÞAR₃, thenhl^?;miforms a plane and hl^?;mi^?¼lVm^?, hence ðm;lÞAR₃. ThereforeR₃is also symmetric.

Finally we show that for anyi;j;kAf0;. . .;4g,

p_ij^k¼ jfnALj ðl;nÞAR_i;ðn;mÞAR_jgj

is independent of the choice of ðl;mÞAR_k. The assertion is clear when k¼0. For

the moment, we put p_ijðl;mÞ ¼ jfnALj ðl;nÞAR_i;ðn;mÞAR_jgj. We can easily see that when ðl;mÞAR_k, p_0jðl;mÞ ¼d_jk. Since each relation is symmetric, p_jiðl;mÞ ¼

p_ijðm;lÞ. SinceR₀;. . .;R₄ form a partition ofLL, we have X⁴

i¼0

p_ii⁰¼ jLj ¼1

2q²ðq1Þ²; and

X⁴

j¼0

pijðl;mÞ ¼ p_ii⁰

for anyiAf0;. . .;4gand for any pairðl;mÞ. Letsbe the permutationð0;2Þð1;3Þon f0;. . .;4g. Then sinceðl;mÞARiif and only ifðl;m^?ÞARsðiÞ,

p_ijðl;mÞ ¼ p_isðjÞðl;m^?Þ ¼ p_sðiÞsðjÞðl;mÞ: ð2Þ Hence we only need to show that p₁₁^k ð1ckc4Þ are independent of the choice of ðl;mÞAR_k.

Lemma 4.2.For1ckc4, p₁₁^k is independent of the choice ofðl;mÞARk and p₁₁⁰ ¼1

2ðq2Þðqþ1Þ²; p¹₁₁¼q²3 2q2;

p²₁₁¼0; p³₁₁¼1

2q²; p₁₁⁴ ¼1

2qðqþ1Þ:

Proof.Fix lAL. Any line which meetsl in a point is in a plane throughl, and con- versely any line in a plane throughlmeetslin a point. Hence by Proposition 4.1 (ii) and (iii),

p₁₁⁰ ¼ jfnALj ðl;nÞAR1gj

¼ X

pAPl

jfnALjnHp;n0lgj

¼ ðqþ1Þ 1

2qðq1Þ 1

¼1

2ðq2Þðqþ1Þ² wherePl:¼ fpAPjlHpg.

Forðl;mÞAR₁, ifnALmeets bothl andm, thennhas a pointlVmornis in the planehl;mi. Hence by Proposition 4.1 (iii)–(v),

p₁₁¹ ¼ jfnALj ðl;nÞ;ðn;mÞAR1gj

¼ jfnALjnJhl;mi;n0l;mgj þ jfnALjlVmAnUhl;migj

¼ 1

2qðq1Þ 2

þ 1

2qðq1Þ 1 2q

¼q²3 2q2:

From (2), we have p₁₁² ¼0. Forðl;mÞAR₃UR₄, we have jfnALj ðl;nÞ;ðn;mÞAR₁gj ¼ X

pAPl

jfnALjmVpAnJpgj:

Ifðl;mÞAR3, then there is just one planep0¼hl;l^?Vmi APlsuch thatp^?₀ ¼mVp0. By Proposition 4.1 (iv), there is no line ofLthroughmVp0and inp0, and for other planep, there areq=2 lines ofLthroughmVpand inp. Hence

p³₁₁¼ jfnALj ðl;nÞ;ðn;mÞAR₁gj

¼ X

pAPlnfp0g

jfnALjmVpAnJpgj

¼q1 2q:

Forðl;mÞAR₄, any planepofP_l hasq=2 lines ofLthroughmVp. So p₁₁⁴ ¼ jfnALj ðl;nÞ;ðn;mÞAR₁gj ¼ ðqþ1Þ 1

2q: r

Therefore ðL;fRig_0cic4Þ becomes a symmetric association scheme. For iA f0;. . .;dg, letB_i:¼ ðp_ij^kÞ_0cj;kc4. ThenB₀ is the identity matrix,

B₁¼

0 1 0 0 0

p⁰₁₁ q²3=2q2 0 q²=2 qðqþ1Þ=2

0 0 0 1 0

0 q²=2 p₁₁⁰ q²3=2q2 qðqþ1Þ=2 0 q²ðq3Þ=2 0 q²ðq3Þ=2 s 0

BB BB B@

1 CC CC CA

;

wheres¼ ðqþ1Þðq²3q2Þ=2,

B₂ ¼

0 0 1 0 0

0 0 0 1 0

1 0 0 0 0

0 1 0 0 0

0 0 0 0 1

0 BB BB B@

1 CC CC CA

;

B3¼

0 0 0 1 0

0 q²=2 p₁₁⁰ q²3=2q2 qðqþ1Þ=2

0 1 0 0 0

p⁰₁₁ q²3=2q2 0 q²=2 qðqþ1Þ=2 0 q²ðq3Þ=2 0 q²ðq3Þ=2 s 0

BB BB B@

1 CC CC CA

;

and

B₄¼

0 0 0 0 1

0 q²ðq3Þ=2 0 q²ðq3Þ=2 s

0 0 0 0 1

0 q²ðq3Þ=2 0 q²ðq3Þ=2 s

p⁰₄₄ qðq3Þðq²3q2Þ=2 p₄₄⁰ s t 0

BB BB B@

1 CC CC CA

;

where p⁰₄₄¼qðq2Þðq3Þðqþ1Þ=2 and t¼qðq3Þðq²3q2Þ=2. The first ei- genmatrix of this association scheme is given by

P¼

1 ðq2Þðqþ1Þ²=2 1 ðq2Þðqþ1Þ²=2 p₄₄⁰ 1 ðq2Þðqþ1Þ=2 1 ðq2Þðqþ1Þ=2 0

1 ðqþ1Þ 1 qþ1 0

1 ðqþ1Þ 1 ðqþ1Þ 2q

1 ðq²3q2Þ=2 1 ðq²3q2Þ=2 qðq3Þ 0

BB BB B@

1 CC CC CA :

5 Proof of Theorem 3.3

In this section, we prove Theorem 3.3 by using Theorem 3.2. The number of vertices of the graphGisjLj=2¼q²ðq1Þ²=4. For a pairfl;l^?gAVG,

ffm;m^?gAVGj fm;m^?gis adjacent tofl;l^?gg

¼ ffm;m^?gAVGjmmeetsl in a pointg

¼ ffm;m^?gAVGj ðl;mÞAR₁g:

So, the size of this set is p₁₁⁰ ¼ ðq2Þðqþ1Þ²=2, which is justkin the definition of strongly regular graph. Next choosefl;l^?g;fm;m^?gAVGwhich are adjacent inG.

We may suppose thatlmeetsmin a point. Then

ffn;n^?gAVGj fn;n^?gis adjacent to bothfl;l^?gandfm;m^?gg

¼ ffn;n^?gAVGjnmeets bothl andmin a pointg Uffn;n^?gAVGjnmeets bothlandm^?in a pointg

¼ ffn;n^?gAVGj ðl;nÞAR₁;ðm;nÞAR₁UR₃g:

Hence the size of this set is p₁₁¹ þp¹₁₃¼ ð3q²3q4Þ=2. This is justlin the defini- tion of strongly regular graph.

Similarly, forfl;l^?g;fm;m^?gAVGwhich are not adjacent inG, sinceðl;mÞAR₄, jffn;n^?gAVGj fn;n^?gis adjacent to bothfl;l^?gandfm;m^?ggj

¼p₁₁⁴ þp⁴₁₃¼qðqþ1Þ:

This is justmin the definition of strongly regular graph.

Alternatively, we can prove Theorem 3.3 by using the quotient association scheme (cf. [1, p. 139, Theorem 9.4]). In the association scheme of Theorem 3.2,R₀UR₂is an equivalence relation onL. So we can define a quotient association scheme on the set of equivalence classesffl;l^?g jlALgwhose relations are

fðfl;l^?g;fm;m^?gÞ j ðl;mÞAR1UR3g ¼the edge set ofG;

fðfl;l^?g;fm;m^?gÞ j ðl;mÞAR₄g;

and the diagonal relation. The first eigenmatrix of this association scheme can be computed fromP(cf. [1, p. 148]):

1 ðq2Þðqþ1Þ²=2 qðq2Þðq3Þðqþ1Þ=4

1 ðqþ1Þ q

1 ðq²3q2Þ=2 qðq3Þ=2

0 B@

1 CA:

The first relation forms a strongly regular graph whose parameters are calculated from the second column of the above first eigenmatrix.

6 Another construction ofG_q

In this section, we will give another construction of the strongly regular graph G_q. This construction uses a method which generalizes a construction of Mathon ([10, p.

137], see also [3, pp. 96–97]).

Let G¼SLð2;qÞ, K¼Oð2;qÞ. Then XðG;KÞis a ðq2Þ=2-class pseudo-cyclic symmetric association scheme (cf. [3, p. 96]). By Lemma 2.1, we can construct a strongly regular graphDðXðG;KÞÞwith parameters

1

4q²ðq1Þ²;1

2ðq2Þðqþ1Þ²;1

2ð3q²3q4Þ;qðqþ1Þ

which are the same as those ofGq. We shall prove that these graphs are isomorphic.

To show this, we use the isomorphism G²FW^þð4;qÞ which maps ðX;YÞ to XnY(see [11, p. 199]). Letl₀be the external line generated byv₁¼^tð0;1;1;0Þ,v₂ ¼

tð1;1;0;aÞ, wherea is an element ofF_q such that the polynomial x²þxþa is irre- ducible overF_q. For an external linel, there are 2ðqþ1Þbasesðu1;u₂Þoflsuch that Qðxu1þyu₂Þ ¼x²þxyþay² for anyx;yAF_q. Indeed, by Witt’s Theorem,K acts regularly on the set of basesðu1;u₂Þofl with the above condition. It follows that the size of this set is equal to jKj ¼2ðqþ1Þ. LetP be the set of nonsingular points in PGð3;qÞand letL¼ flUl^?jlALg. Then the following lemma holds.

Lemma 6.1. The group W^þð4;qÞ ¼ fXnYjX;YAGg is flag-transitive on the inci- dence structureðP;L;AÞ.Under the isomorphism G²FW^þð4;qÞ,the groups DðGÞ;K² are the stabilizers of an element ofP;L,respectively.

Proof.LetX¼ ðxijÞ_1ci;jc2,Y¼ ðy_ijÞ_1ci;jc2AG. Since

XnY¼

x11y11 x11y12 x12y11 x12y12

x11y21 x11y22 x12y21 x12y22

x21y11 x21y12 x22y11 x22y12

x₂₁y₂₁ x₂₁y₂₂ x₂₂y₂₁ x₂₂y₂₂ 0

BB B@

1 CC CA;

XnY fixesv₁ if and only if

x11y12þx12y11 ¼x21y22þx22y21 ¼0;

x11y22þx12y21 ¼x21y12þx22y11 ¼1:

This implies

W^þð4;qÞ_v1 ¼ fXnXjXAGgFDðGÞ: ð3Þ ForX AG,XnX fixesv2 if and only if

x₁₁² þx11x12þax₁₂² ¼1;

x11x21þx12x21þax12x22 ¼0;

x₂₁² þx21x22þax₂₂² ¼a:

From these, we have

W^þð4;qÞ_v1;v2¼ XnXjX¼ a b ab aþb

AG

which is of orderqþ1. Hence

jfðMv1;Mv2Þ jMAW^þð4;qÞgj ¼ jW^þð4;qÞj=ðqþ1Þ

¼q²ðq1Þ²ðqþ1Þ:

SinceQðxv1þyv₂Þ ¼x²þxyþay²for anyx;yAF_q,

jfðu1;u₂Þ jQðxu1þyu₂Þ ¼x²þxyþay²for allx;yAF_qgj

¼ jLj 2ðqþ1Þ

¼q²ðq1Þ²ðqþ1Þ:

Hence W^þð4;qÞ acts transitively on the set of pairs ðu1;u2Þ such that Qðxu1þyu2Þ ¼x²þxyþay² for any x;yAFq. In particular, W^þð4;qÞ is flag- transitive onðP;L;AÞ.

The equality (3) means that the stabilizer ofhv1i APis isomorphic toDðGÞ. Let A:¼ 1 0

1 1

; B:¼ a0 b0

ab0 a0þb0

AG

such that B is of order qþ1. Then the group hA;Bi is isomorphic to K. AnI, InA interchange l₀ andl₀^?, while BnI,InBfixl₀ andl₀^?. So fXnYjX;Y A hA;Bigis a subgroup ofW^þð4;qÞ_l0Ul₀^?. SinceW^þð4;qÞ_l0Ul₀^? has order 4ðqþ1Þ²¼ jKj², we have thatW^þð4;qÞ_l0Ul₀^? is isomorphic toK². r Theorem 6.2.The graphGqis isomorphic toDðXðSLð2;qÞ;Oð2;qÞÞÞ.

Proof. The graph Gq is isomorphic to the collinearity graph of the dual of the inci- dence structureðP;L;AÞ. From Lemma 6.1, the dual ofðP;L;AÞis isomorphic to the coset geometryðG²=K²;G²=DðGÞ;Þdefined in Lemma 2.2. From Lemma 2.2, the collinearity graph ofðG²=K²;G²=DðGÞ;Þis isomorphic to DðXðG;KÞÞ. There-

foreGq is isomorphic toDðXðG;KÞÞ. r

References

[1] E. Bannai, T. Ito,Algebraic combinatorics. I.Benjamin/Cummings Publishing Co. 1984.

MR 87m:05001 Zbl 0555.05019

[2] A. E. Brouwer, R. Mathon, unpublished.

[3] A. E. Brouwer, J. H. van Lint, Strongly regular graphs and partial geometries. In:Enu- meration and design(Waterloo, Ont.,1982), 85–122, Academic Press 1984. MR 87c:05033 Zbl 0555.05016

[4] P. Dembowski,Finite geometries. Springer 1968. MR 38 #1597 Zbl 0159.50001

[5] G. L. Ebert, S. Egner, H. D. L. Hollmann, Q. Xiang, A four-class association scheme.

J. Combin. Theory Ser. A96(2001), 180–191. MR 2002j:05152 Zbl 1010.05083 [6] C. D. Godsil, Association schemes. Preprint.

[7] C. D. Godsil, private communication.

[8] J. W. P. Hirschfeld,Finite projective spaces of three dimensions. Oxford Univ. Press 1985.

MR 87j:51013 Zbl 0574.51001

[9] M. Koppinen, On algebras with two multiplications, including Hopf algebras and Bose- Mesner algebras.J. Algebra182(1996), 256–273. MR 97e:16092 Zbl 0897.16022

[10] R. Mathon, 3-class association schemes. In: Proceedings of the Conference on Algebraic Aspects of Combinatorics(Univ. Toronto, Toronto, Ont.,1975), 123–155. Congressus Nu- merantium, No. XIII, Utilitas Math., Winnipeg, Man. 1975. MR 54 #2503

Zbl 0326.05023

[11] D. E. Taylor,The geometry of the classical groups. Heldermann 1992. MR 94d:20028 Zbl 0767.20001

Received 3 June, 2002

T. Fujisaki, Graduate school of Mathematics, Kyushu University, 6-10-1 Hakozaki, Fukuoka 812-8581, Japan

Email: [email protected]