Dam, E.R. van

English

Research Papers in Economics

Designs, Codes and Cryptography, 21, 83–86, 2000.
c ° 2000 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
A Characterization of Association
Schemes from Afﬁne Spaces
EDWIN R. VAN DAM Edwin.vanDam@kub.nl
Tilburg University, Dept. Econometrics, PO Box 90153, 5000 LE Tilburg, The Netherlands
Abstract. We characterize the association schemes from afﬁne spaces as the association schemes in which all
relations are equivalence relations (when united with the identity relation). The schemes from afﬁne spaces of
dimensionatleastthreearecounterexamplesofaconjectureofA.V.Ivanov[8,Problem1.3]onamorphicschemes.
Keywords: Afﬁne geometry, association schemes
1. Introduction
In this note we assume the reader has some basic knowledge of afﬁne spaces and asso-
ciation schemes. For some background on afﬁne spaces we refer the reader to [3], for
some background on association schemes to [2]. Furthermore, we shall use the following
terminology.
The irreﬂexive part of a relation R ½ V £ V is deﬁned as the relation R n I, where I is
the identity relation on V, i.e in the irreﬂexive part of a relation an element is not related
to itself. An equivalence relation is called uniform if all its equivalence classes have the
same size. When seen as a graph, the irreﬂexive part of such a relation is a disjoint union
of cliques of the same size, hence it is a strongly regular graph (or a complete graph).
In a ﬁnite afﬁne space each parallel class of lines naturally deﬁnes a (uniform) equivalence
relation on the points of the afﬁne space. Moreover, the irreﬂexive parts of the equivalence
relations coming from all parallel classes in the afﬁne space form the set of non-identity
relations of an association scheme. We shall give a characterization of these association
schemes by proving that they are the only association schemes in which every non-identity
relation is the irreﬂexive part of an equivalence relation. This answers the question which
association schemes can be “built” with the most elementary relations: equivalence rela-
tions.
For translation schemes the characterization follows from a result of Baer [1] (see also
[9, Section 29]) on so-called geometric partitions of Abelian groups. Note that by the
Veblen-Young theorem (cf. [3]) a ﬁnite afﬁne space is a ﬁnite afﬁne plane (not necessarily
Desarguesian) or AG.n;q/, the n-dimensional afﬁne geometry (vector space) over GF.q/,
hence (of the afﬁne spaces) only the afﬁne planes which are not translation planes give rise
to association schemes that are not translation schemes. Note however that the restriction
to translation schemes is a rather strong one. Other related results have been obtained
by Cameron [4, Thm. 2B.1], who showed that a .v ¡ 1/-class association scheme on v
vertices must be the association scheme from the lines of AG.n;2/, for some n. This84 VAN DAM
result is implied by a result of Ferguson and Turull [6], who showed that a—not necessarily
symmetric—association scheme with all valencies equal to 1 is the direct product of cyclic
association schemes of prime power order (and they also give some other characterizations
of these association schemes). Rao, Ray-Chaudhuri and Singhi [10] characterize all—not
necessarilysymmetric—associationschemesforwhichtheintersectionparameters pk
ij · 1,
for all i; j;k 6D 0. These are precisely the association schemes with all valencies 1 and,
if the number of vertices is odd, their symmetrized schemes (a symmetrized scheme is
obtained by uniting each relation with its transpose).
Note that the association schemes that we shall characterize are precisely the association
schemes for which the intersection parameters pk
ii D 0 for all i;k 6D 0, i 6D k.W ed on o t
even have to restrict to symmetric association schemes, since a relation that is uniform and
transitive is necessarily symmetric.
2. The Characterization
Supposewehaveanassociationschemewithvertexset V andrelations Ri, i D 0;1;:::;d,
withcorrespondingadjacencymatrices Ai,i D 0;1;:::;d,where R0 istheidentityrelation.
Assume that Ri [ R0 is an equivalence relation for all i 6D 0. Then on the point set V we
deﬁne lines as the equivalence classes from all these equivalence relations. Note that this
gives a linear space, i.e. through any two points there is precisely one line, and lines have
size at least two.
LEMMA Three noncollinear points lie in a subspace which is an afﬁne plane.
Proof. Consider three noncollinear points x, y and z. Without loss of generality x and y
are on a line from relation R1, and x and z are on a line from relation R2. Now consider the
set X of points p that are on a line from relation R2 that intersects the line through x and y.
Algebraically this deﬁnition is expressed by .A1 C I/.A2 C I/xp > 0. Note that x, y and z
are elements of X.
Since A1 and A2 commute, it follows that X and the lines from relations R1 and R2
through X form a grid. This implies that for all u 2 X, .A1 C I/.A2 C I/uw > 0i fa n d
only if w 2 X.
Now take two points u and w from X. We want to show that the entire line through u and
w is contained in X.I fu and w are on a line from relations R1 or R2, then this is clearly
the case. Next, suppose that u and w are not on a line from relations R1 and R2 (u and w
are noncollinear on the grid). Without loss of generality we may assume that u and w are
on a line from relation R3, hence .A3/uw D 1. It is also clear that .A1A2/uw D 1, and so
it follows from the equation A1A2 D
P
k pk
12Ak that the intersection parameter p3
12 equals
1. From the construction of X, and the equation A1A2 D
P
k pk
12Ak it now follows that all
vertices v such that .A3/uv D 1 are contained in X. Hence also in this case the line through
u and w is contained in X, and we proved that X is a subspace.
Next, consider a line l in X, and a point p in X, not on l. To show that X is an afﬁne
plane, it sufﬁces to show that there is a line in X containing p and parallel to l.I fl is from
relation R1 or R2, then this is obviously the case. If the line l is from relation Ri, i 6D 1;2,ASSOCIATION SCHEMES FROM AFFINE SPACES 85
then it follows that pi
12 > 0, and hence that the line through p from relation Ri is also
contained in X. Thus in all cases there is a line in the subspace X through p, parallel to l,
which proves that X is an afﬁne plane.
THEOREM Let .V;fRigiD0;1;:::;d/ be a d-class association scheme, d > 1, such that Ri [ R0
is an equivalence relation for all i 6D 0. Then the set of all equivalence classes of these
equivalence relations is the set of lines in an afﬁne space on point set V. In particular, all
non-identity relations are isomorphic.
Proof. As mentioned before, the set of lines on point set V is a linear space. Moreover, by
thelemmawehavethatanythreenoncollinearpointsarecontainedinasubspacewhichisan
afﬁne plane. Since parallelism in these planes is an equivalence relation (this follows since
if lines are parallel, then they are from the same relation; and each relation is by assumption
(the irreﬂexive part of) an equivalence relation), and there exist three noncollinear points
(since d > 1), it follows that the linear space is an afﬁne space (see for example Thm. 2.7
of [3]).
Theproofofthetheoremusesthefactthatwehaveanassociationscheme. Itisnotpossible
toonlyusethefactthatwehaveapartitionofthecompleterelationintotheidentityrelation
and the irreﬂexive parts of uniform equivalence relations, such that the adjacency matrices
of any two relations commute (the union of any two relations is a disjoint union of grids),
sincetherearesuch“schemes”thatarenotassociationschemes. Forexample, in AG.n;q/,
n ¸ 3,onemighttakeoneparallelclassofplanesforonerelation,andtheparallelclassesof
lineswhicharetransversaltotheseplanesfortheotherrelations;hereallrelationscommute,
but this is not an association scheme (for example, by the theorem). In fact, what we found
here are counterexamples of a conjecture on amorphic association schemes.
3. A False Conjecture on Amorphic Association Schemes
An association scheme is called amorphic if any way of fusing the non-identity relations
againgivesanassociationscheme. Itisprovenin[7]thatinanamorphicd-classassociation
scheme with d ¸ 3 all non-identity relations are strongly regular graphs of Latin square
typeorstronglyregulargraphsofnegativeLatinsquaretype. Theassociationschemesfrom
afﬁne planes are amorphic of the ﬁrst type. (For more on amorphic association schemes
we refer to [5].)
A.V. Ivanov [8, Problem 1.3] conjectured that an association scheme in which all non-
identity relations are strongly regular graphs must be amorphic. The association schemes
from AG.n;q/, n ¸ 3, are clearly counterexamples of this conjecture by the remarks
just before this section (the parallel class of planes is a fusion of parallel classes of lines).
Note also that the non-identity relations in these schemes are not of Latin square type or
negativeLatinsquaretype. AccordingtoM.Klin[privatecommunication],Ivanovprobably
intendedtoconjecturethataprimitiveassociationschemeinwhichallnon-identityrelations
are strongly regular graphs must be amorphic. This weaker conjecture still stands.86 VAN DAM
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291.

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