Caen, D. de

Dam, E.R. van

English

Research Papers in Economics

doi:10.1006/eujc.2000.0467
Available online at http://www.idealibrary.com on
Europ. J. Combinatorics (2001) 22, 297–301
Fissioned Triangular Schemes via the Cross-ratio
D. DE CAEN AND E. R. VAN DAM
A construction of association schemes is presented; these are ﬁssion schemes of the triangular
schemes T.n/ where n D q C 1 with q any prime power. The key observation is quite elementary,
being that the natural action of PGL.2;q/ on the 2-element subsets of the projective line PG.1;q/
is generously transitive. In addition, some observations on the intersection parameters and fusion
schemes of these association schemes are made.
c  2001 Academic Press
1. THE CONSTRUCTION
This paper is a sequel to [4]. In that paper, it was observed that almost all known self-
dual classical association schemes have natural ﬁssion schemes (ﬁssioning the maximum-
distance relation only); whereas in the non-self-dual case there seemed to be no analogous
ﬁssion schemes. Subsequently, we found that there is at least one such non-self-dual clas-
sical association scheme that admits an interesting ﬁssion scheme, namely the triangular
scheme T.n/ D J.n;2/ where n D q C 1 with q any prime power; this is the object of
the present work. For terminology and background, we refer to Bannai and Ito [2] for asso-
ciation schemes and to Hirschfeld [9] for ﬁnite geometry. Recall that the group PGL.2;q/
acts (as M¨ obius transformations) on the projective line PG.1;q/; this action is (sharply) 3-
transitive. There is a natural induced action on the 2-element subsets of the projective line,
namely M.fx; yg/ VD fM.x/; M.y/g for each M in PGL.2;q/. In the proof below we apply
the basic fact (cf. [9, p. 135]) that the cross-ratio
.a;b;c;d/ VD
.a   c/.b   d/
.a   d/.b   c/
is a complete invariant for ordered quadruples of distinct points on the projective line, i.e., one
quadruple may be mapped to another quadruple (via a M¨ obius transformation) if and only if
they have the same cross-ratio.
THEOREM. The action of PGL.2;q/ on the 2-element subsets of PG.1;q/ is generously
transitive.
PROOF. Given intersecting 2-sets fa;bg and fa;cg, there is some M in PGL.2;q/ that
swaps them, since the group is triply transitive. And given disjoint 2-sets fa;bg and fc;dg,
there is also some M¨ obius transformation that interchanges them, because the ordered quadru-
ples .a;b;c;d/ and .c;d;a;b/ have the same cross-ratio. 2
Given any transitive permutation group G acting on a set , the orbitals are the orbits in
   under the natural action of G on pairs. If G is generously transitive, then the orbitals
form the relations (associate classes) of a symmetric association scheme (cf. [2, p. 54]). In
our case, the relations can be described as follows. One relation, say R1, is the line-graph
of the complete graph (i.e., one relation of the triangular scheme T.q C 1/ has remained
unﬁssioned). Next, for each reciprocal pair fs;s 1g of elements in GF.q/nf0;1g, there is a
relation Rfs;s 1g where fa;bg and fc;dg are in this relation when .a;b;c;d/ equals s or s 1.
Note that .b;a;c;d/ D .a;b;c;d/ 1 so this makes sense as a deﬁnition for unordered
0195–6698/01/030297 + 05 $35.00/0 c  2001 Academic Press298 D. de Caen and E. R. van Dam
pairs fa;bg. Henceforth we will write Rs instead of Rfs;s 1g for typographical reasons; note
that since the ﬁeld element 1 cannot occur as a cross-ratio, this notation will not conﬂict with
that of relation R1 above.
We now easily ﬁnd that this ﬁssioned triangular scheme, which we shall denote by
FT.q C 1/, has 1
2.q C 1/ associate classes if q is odd and 1
2q classes if q is even. When
q is odd the ﬁeld element  1 is equal to its own reciprocal; thus the relation R 1 has valency
1
2.q   1/ which is half the valency of the other relations Rs with s in GF.q/nf0;1; 1g. The
relation R1 has valency 2.q   1/.
We remark that for small odd q the relation R 1 is a familiar object: for q D 5 it is the
line-graph of Petersen’s graph; for q D 7 it is the Coxeter graph (this was apparently known
to Coxeter himself, cf. [6, p. 122]); for q D 9 it is the line-graph of Tutte’s 8-cage. There
seem to be some other such ‘sporadic isomorphisms’: for example when q D 11 the relation
R2 D Rf2;6g is the line-graph of the point-block incidence graph of the (unique) symmetric
.11;6;3/-design; and when q D 9 and fs;s 1g is the pair of primitive fourth roots of unity,
then Rs is the second subconstituent of the Gewirtz graph (cf. [5, p. 106]).
2. INTERSECTION PARAMETERS
It is possible to give explicit formulas for the intersection parameters pk
ij of the association
scheme FT.q C 1/; we now sketch the main points of the derivation. The cases q odd and q
evenaresimilar,withthelattercasebeingslightlycleanersincetheexceptionalcase‘ D  1’
does not occur. Hence we will only present the case q even; besides, this case is the more
pertinent one in the discussion of fusion schemes in Section 3.
Thus let q D 2e be any power of two. The scheme FT.2eC1/ has 2e 1 classes. The relation
R1 has valency 2.q  1/ and each of the other relations Rs D Rfs;s 1g (for s in GF.q/nf0;1g)
has valency q   1. The intersection parameters involving R1 are easy to work out and we
list them without proof: for distinct r and s (and s 6D r 1) in GF.q/nf0;1g, p1
11 D q   1,
pr
11 D 4, p1
1r D 2, p1
rr D 1, and p1
rs D 2.
Now let the symbols r, s and t represent three (not necessarily distinct) elements of
GF.q/nf0;1g; we aim at a formula for pr
st. What one has to do is ﬁx a pair of 2-sets fa;bg
and fc;dg in relation Rr, and count the number of 2-sets fx; yg such that fa;bg and fx; yg are
in relation Rs and fc;dg and fx; yg are in relation Rt. The triple transitivity of PGL.2;q/ is
useful here, since it implies that we may take, without loss of generality, fa;bg D f1;0g and
fc;dg D f1;rg. For the unknown pair fx; yg we then obtain the two equations
s or s 1 D
.1   x/.0   y/
.1   y/.0   x/
D
y
x
(1)
and
t or t 1 D
.1   x/.r   y/
.1   y/.r   x/
: (2)
Equations (1) and (2) together involve two essentially different cases, not four, since fy;xg D
fx; yg; thus we may ﬁx the left-hand side of (1) as being s, and examine the two cases for (2)
in turn. In the ﬁrst case we have y D sx and
t D
.1   x/.r   y/
.1   y/.r   x/
D
.1   x/.r   sx/
.1   sx/.r   x/
:
This leads to the following quadratic for x (after changing all minus signs to plus signs, as we
may since we are in characteristic two):
s.t C 1/x2 C .rst C r C s C t/x C r.t C 1/ D 0: (3)Fissioned triangular schemes via the cross-ratio 299
The other case (when the left-hand side of (2) is t 1) leads to the similar quadratic
s.t C 1/x2 C .rs C rt C st C 1/x C r.t C 1/ D 0: (4)
Note that since r;s and t are all in GF.q/nf0;1g, Eqns (3) and (4) are genuine quadratics,
with non-zero quadratic and constant terms. The linear coefﬁcient .rstCrCsCt/ in (3) could
equal 0, in which case the unique solution for x is the square root of r
s. If rst Cr Cs Ct 6D 0,
then (3) has (two) solutions x if and only if
Tr

rs.t C 1/2
.rst C r C s C t/2

D 0 (5)
where Tr.z/ is the trace map from GF.2e/ onto GF.2/. Similarly, if rs C rt C st C 1 6D 0,
then (4) has (two) solutions x if and only if
Tr

rs.t C 1/2
.rs C rt C st C 1/2

D 0: (6)
Thus pr
st has a value of anywhere from 0 to 4. A reasonably concise formula is the fol-
lowing: let A D A.r;s;t/ be the expression for the argument of the trace map in (5), and
B D B.r;s;t/ be the one for (6). Then, when rst C r C s C t 6D 0 and rs C rt C st C 1 6D 0
pr
st D 2 C . 1/TrTAU C . 1/TrTBU (7)
with the obvious modiﬁcations being made in the other cases. Incidentally, it is easy to check
that .rst C r C s C t/ and .rs C rt C st C 1/ cannot simultaneously equal 0.
We make one more remark concerning the form of the intersection parameters. The ex-
pressions A.r;s;t/ and B.r;s;t/ are not symmetric in s and t, hence formula (7) for pr
st
appears not to be symmetric either. This may seem strange, since we know from general prin-
ciples that pr
st D pr
ts. An explanation for this is the following. A.r;s;t/ has the same trace
as C.r;s;t/ VD rsCrtCst
.rstCrCsCt/2 since their sum is of the form
xy
x2Cy2 and such ﬁeld elements, in
characteristic two, must have trace 0 (exercise for the reader).
Similarly, B.r;s;t/ has the same trace as D.r;s;t/ VD rst.rCsCt/
.rsCrtCstC1/2. Thus we may replace
A by C and B by D in (7) without changing the value of the right-hand side; and C and
D are both symmetric functions of the three variables r;s and t. This conﬁrms the fact that,
since the valencies nr are the same for all r in GF.q/nf0;1g, the intersection parameter pr
st
is symmetric in all three variables.
It would be interesting to ﬁnd explicit formulas for the entries of the eigenmatrix (character
table) of FT.qC1/. One strategy for doing this (used by Bannai and his co-workers in several
papers; see [1] for a survey) is the following. First calculate all of the intersection parameters;
it is usually feasible to do this, at least in some reasonable algebraic form perhaps involving
character sums. This tells us what the intersection matrices Bi.k; j/ VD pk
ij are. Secondly,
from these Bi’s (at small values of q) it may be possible to guess what the eigenmatrix P
should be. Once the right guess has been made it is usually straightforward to actually prove
the result, using Theorem II.4.1 in [2]. Unfortunately, we have been unable so far to guess the
general shape of P for our schemes FT.q C 1/; we generated these character tables using a
computer for all prime powers q less than 40, and they seem to have a very complicated form.
3. FUSION SCHEMES
Given any association scheme, it is of interest to determine all of its fusion schemes (also
called subschemes). This is in general a very hard problem that has not been worked out300 D. de Caen and E. R. van Dam
completely even for quite classical examples such as the Johnson schemes (cf. [10]). In the
case of the schemes FT.q C 1/, there is of course the original two-class triangular scheme
T.qC1/. Observe also that if q D pe is a proper power of p, then the Frobenius map x 7! x p
(and its iterates) gives a fusion scheme. In other words, there is an overgroup (P0L.2;q/ in
case p is prime) of PGL.2;q/, and the orbitals under this overgroup constitute a fusion
scheme of FT.q C 1/.
Limited computational evidence suggests that FT.q C 1/ has no other non-trivial fusions,
except maybe in some sporadic cases, and when q D 4 f ( f any integer at least 2) where
there seems to be an interesting 4-class fusion scheme. We say ‘seems’ because we are lack-
ing a proof that this is indeed an association scheme. To describe this (putative) scheme, let
the ground-set be all 2-element subsets of the projective line PG.1;4 f /; the four possible
relations for two distinct 2-sets fa;bg and fc;dg are:
S1: fa;bg \ fc;dg 6D ;, i.e., R1 in the earlier notation.
S2: fa;bg\fc;dg D ; and the cross-ratio  D .a;b;c;d/ satisﬁes 2 f  1 D 1, i.e.,  lies
in the subﬁeld GF.2 f /.
S3: fa;bg \ fc;dg D ; and the cross-ratio  D .a;b;c;d/ satisﬁes 2 f C1 D 1.
S4: The remainder.
We have been able to show by computer that these four relations do indeed form a scheme
when f is less than or equal to 6. In addition, we can prove in general that some of the
intersection parameters, such as p3
23, are well deﬁned; but certain other parameters such as
p3
33 have left us bafﬂed. An explicit knowledge of the eigenmatrix of FT.4 f C 1/ would
theoretically settle this question (cf. [10, Lemma 1]), which is partly why we raised the issue
of computing it earlier.
CONJECTURE. The above relations Si on the 2-subsets of PG.1;4 f / do form a 4-class
association scheme for all f  2. The corresponding eigenmatrix is given by
P D
2
6 6
6
4
1 2.4 f   1/ .2 f  1   1/.4 f   1/ 2 f  1.4 f   1/ 2 f .2 f  1   1/.4 f   1/
1 4 f   3 2   2 f  2 f  2 f .2 f   2/
1  2 1   2 f 0 2 f
1  2 .2 f  1   1/.2 f   1/ 2 f  1.2 f   1/  2 f .2 f   2/
1  2 2 f  1.2 f   1/ C 1  2 f  1.2 f C 1/ 2 f
3
7 7
7
5
:
We note ﬁnally that, granting this conjecture, one can merge S2 and S3 to obtain a 3-class
scheme, and then further merge S1 with S2 and S3 to obtain a 2-class scheme. The resulting
graph G D S1 [ S2 [ S3 is strongly regular with parameters v D 22 f  1.22 f C 1/, k D
.2 f C1/.22 f  1/,  D .2 f  1/.32 f C2/,  D 2 f C1.2 f C1/. Graphs with these parameters
have already been constructed by Brouwer and Wilbrink (cf. [3, 7B]); it was checked that in
the smallest case f D 2 .v D 136/ the two constructions yield isomorphic strongly regular
graphs. We know nothing for larger values; but the two constructions look totally different, so
that it is a reasonable guess that they are not isomorphic in general.
ADDED IN PROOF. The above conjecture is proven by Tanaka [11] and independently by
Ebert, Egner, Hollmann, and Xiang [7,8]. Tanaka gives a group theoretic proof using charac-
ters, while Ebert et al. give a geometric proof using inversive planes in [7], and a direct proof
from the intersection parameters in [8]. In [7] it is also proved that the strongly regular graph
G D S1 [ S2 [ S3 is isomorphic to the Brouwer-Wilbrink strongly regular graph.Fissioned triangular schemes via the cross-ratio 301
REFERENCES
1. E. Bannai, Character tables of commutative association schemes, in: Finite Geometries, Buildings,
and Related Topics, Oxford University Press, Oxford, 1990, pp. 105–128.
2. E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings Publ.,
San Francisco, 1984.
3. A. E. Brouwer and J. H. van Lint, Strongly regular graphs and partial geometries, in: Enumeration
and Design, Academic Press, New York, 1984, pp. 85–122.
4. D. de Caen and E. R. van Dam, Fissions of classical self-dual association schemes, J Comb Theory,
Ser. A, 88 (1999), 167–175.
5. P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and their Links, Cambridge University
Press, Cambridge, 1991.
6. H. S. M. Coxeter, My graph, Proc. London Math. Soc., 46 (1983), 117–136.
7. G. L. Ebert, S. Egner, H. D. L. Hollmann and Q. Xiang, On a four-class association scheme,
preprint.
8. G. L. Ebert, S. Egner, H. D. L. Hollmann and Q. Xiang, Proof of a conjecture by de Caen and van
Dam, preprint.
9. J. W. P. Hirschfeld, Projective Geometry Over Finite Fields, 2nd edn, Oxford University Press,
Oxford, 1998.
10. M. E. Muzychuk, Subschemes of the Johnson scheme, Europ. J. Combinatorics, 13 (1992), 187–
193.
11. H. Tanaka, A 4-class subscheme of the association scheme coming from the action of PGL.2;4 f /,
preprint.
Received 6 October 1999 and accepted 3 August 2000
D. DE CAEN
Department of Mathematics and Statistics,
Queen’s University,
Kingston,
Canada K7L 3N6
E-mail: decaen@mast.queensu.ca
AND
E. R. VAN DAM
Department of Econometrics and Operations Research,
Tilburg University,
P.O. Box 90153, 5000 LE Tilburg,
The Netherlands
E-mail: Edwin.vanDam@kub.nl

Fissioned triangular schemes with the cross-ratio.

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics