Association scheme
Association schemes were introduced by R.C. Bose and T. Shimamoto [a4], studied further via the Bose–Mesner algebra introduced in [a3], generalized and given a most important impetus by P. Delsarte [a8], and generalized further by D.G. Higman [a10], [a11], [a12] to the theory of coherent configurations. The first text devoted to the theory is [a1]. A recent text that develops the theory both quite generally and quite extensively is [a9].
Association schemes provide the appropriate setting for treating certain problems from several different areas of algebraic combinatorics, for example, coding theory, design theory, algebraic graph theory, finite group theory, and finite geometry. The following definition is equivalent to that of Delsarte (1973).
An association scheme with
classes is a finite set
together with
relations
on
such that:
i) is a partition of
;
ii) ;
iii) For each there is a unique
for which
if and only if
; that is,
;
iv) for any , the number
of
with
and
depends only on
,
,
;
v) for all
.
The numbers are called the intersection numbers of the association scheme. For
, put
, and
. Note that for each
for which
,
is a simple graph which is regular of degree
. Indeed, a pair of complementary strongly regular graphs is equivalent to an association scheme with two classes.
The Bose–Mesner algebra is the matrix algebra generated by the adjacency matrices
of
, where
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The matrices sum to
(the matrix whose every entry is
),
, and
, for all
. The matrices
are linearly independent and generate a commutative
-dimensional semi-simple algebra
with a unique basis of minimal idempotents
. The interplay between the two bases for
leads to useful restrictions on the various parameters, for example the so-called Krein conditions (discovered by L.L. Scott Jr., 1973). These restrictions are given as follows: Write
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Then the Krein parameters are all non-negative.
The matrix is a minimal idempotent, and one may take
. Let
and
be the matrices relating the two bases for
:
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It follows that , which implies that the
are eigenvalues of
, and the columns of
are the corresponding eigenvectors. Thus,
is the multiplicity of the eigenvalue
of
. The fact that the
must all be integers is known as the rationality condition for an association scheme.
Delsarte's linear programming bound.
Let be a non-empty subset of the underlying set
for a symmetric association scheme. Define the inner distribution
of
by
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So, is the average number of points in
in relation
with a fixed point of
. Then Delsarte's linear programming bound says that
. This set of inequalities has been used to obtain upper bounds on the size of cliques and lower bounds on the size of designs in various structures.
Examples.
Two of the most studied association schemes are the Johnson and Hamming schemes. The Johnson scheme has for its set
the set of all
-subsets of a set of size
. Then
. The Hamming scheme
has the set of all words of length
over an alphabet of
symbols as its set
. The relation
consists of all pairs
for which the Hamming distance (cf. also Error-correcting code) between
and
is
.
The finite generalized -gons with parameters
introduced by J. Tits (see [a15]) provide examples of strongly regular graphs to which the theory of association schemes may be applied. The rationality conditions can be used to show that
. The Krein conditions force
for
or
, and
for
. For a different type of application of the theory of association schemes to finite generalized
-gons (i.e., generalized quadrangles, cf. also Quadrangle; Quadrangle, complete), see [a13].
An association scheme with just two classes is equivalent to a pair of complementary strongly regular graphs. The theory of strongly regular graphs is subsumed in the theory of distance-regular graphs. Start with a connected simple graph with vertex set
of diameter
. Define
by
whenever
and
are at distance
in
. When this defines an association scheme, the graph
is called distance-regular. Many of the association schemes that arise in combinatorics are of this type. The major reference for distance-regular graphs is [a6]. A more modest introduction to the subject is [a2]. See [a5] for a table of strongly regular graphs with at most
vertices. For many applications to coding theory see [a8], [a14], [a7].
Let be a permutation group acting on a set
. Then
has a natural action on
whose orbits are known as orbitals. If for each pair
of distinct elements of
there is an element of
interchanging
and
, then the orbitals form an association scheme. The Bose–Mesner algebra in this case is known as the centralizer algebra, and the standard results (as given in [a18], for example) have their analogues for the Bose–Mesner algebra. With no special hypotheses on the permutation group
a coherent configuration is obtained, a natural motivation for the work of Higman [a10], [a11], [a12].
P. Terwilliger (see [a16] and its references) has pushed the theory of association schemes much further, and in [a17] has extended the Bose–Mesner algebra to what is now sometimes called the Terwilliger algebra.
References
[a1] | E. Bannai, T. Ito, "Algebraic combinatorics I: association schemes" , Lecture Notes , 58 , Benjamin-Cummings (1984) |
[a2] | N.L. Biggs, "Algebraic graph theory" , Tracts in Math. , 67 , Cambridge Univ. Press (1974) |
[a3] | R.C. Bose, D.M. Mesner, "On linear associative algebras corresponding to association schemes of partially balanced designs" Ann. Math. Stat. , 30 (1959) pp. 21–38 |
[a4] | R.C. Bose, T. Shimamoto, "Classification and analysis of partially balanced incomplete block designs with two associate classes" J. Amer. Statist. Assoc. , 47 (1952) pp. 151–184 |
[a5] | A.E. Brouwer, "Strongly regular graphs" C.J. Colbourn (ed.) J.H. Dinitz (ed.) , The CRC Handbook of Combinatorial Designs , CRC (1996) pp. Part VI, Chapt. 5 |
[a6] | A.E. Brouwer, A.M. Cohen, A. Neumaier, "Distance-regular graphs" , Ergebn. Math. (3) , 18 , Springer (1989) |
[a7] | P.J. Cameron, J.H. van Lint, "Designs, graphs, codes and their links" , London Math. Soc. Student Texts , 22 , Cambridge Univ. Press (1991) |
[a8] | Ph. Delsarte, "An algebraic approach to the association schemes of coding theory" Philips Research Reports Suppl. , 10 (1973) |
[a9] | C.D. Godsil, "Algebraic combinatorics" , Chapman&Hall (1993) |
[a10] | D.G. Higman, "Invariant relations, coherent configurations and generalized polygons" M. Hall Jr. (ed.) J.H. van Lint (ed.) , Combinatorics , Math. Centre Tracts , 57 , Reidel (1975) pp. 347–363 |
[a11] | D.G. Higman, "Coherent configurations, Part I: Ordinary representation theory" Geom. Dedicata , 4 (1975) pp. 1–32 |
[a12] | D.G. Higman, "Coherent configurations, Part II: Weights" Geom. Dedicata , 5 (1976) pp. 413–424 |
[a13] | S. Hobart, S.E. Payne, "Reconstructing a generalized quadrangle from its distance two association scheme" J. Algebraic Combin. , 2 (1993) pp. 261–266 |
[a14] | F.J. MacWilliams, N.J.A. Sloane, "The theory of error–correcting codes" , North-Holland (1977) |
[a15] | S.E. Payne, J.A. Thas, "Finite generalized quadrangles" , Pitman (1984) |
[a16] | P. Terwilliger, "The subconstituent algebra of an association scheme (part III)" J. Algebraic Combin. , 2 (1993) pp. 177–103 |
[a17] | P. Terwilliger, "Algebraic graph theory" Notes, January (1994) |
[a18] | H. Wielandt, "Finite permutation groups" , Acad. Press (1964) |
Association scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Association_scheme&oldid=42839