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The theory of '''association schemes''' arose in [[statistics]], in the theory of [[design of experiments|experimental design]] for the [[analysis of variance]].<ref>{{harvnb|Bailey|2004|loc=pg. 387}}</ref><ref>{{harvnb|Bose|Mesner|1959}}</ref><ref>{{harvnb|Bose|Nair|1939}}</ref> In [[mathematics]], association schemes belong to both [[algebra]] and [[combinatorics]]. In [[algebraic combinatorics]], association schemes provide a unified approach to many topics, for example [[combinatorial design]]s and [[coding theory|the theory of error-correcting codes]].<ref>{{harvnb|Bannai|Ito|1984}}</ref><ref>{{harvnb|Godsil|1993}}</ref> In algebra, association schemes generalize [[group (mathematics)|groups]], and the theory of association schemes generalizes the [[group character|character theory]] of [[group representation|linear representations of groups]].<ref>{{harvnb|Bailey|2004|loc=pg. 387}}</ref><ref>{{harvnb|Zieschang|2005b}}</ref><ref>{{harvnb|Zieschang|2005a}}</ref>

== Definition ==

An ''n''-class association scheme consists of a [[Set (mathematics)|set]] ''X''  together with a [[partition of a set|partition]] ''S'' of ''X'' × ''X'' into ''n''&nbsp;+&thinsp;1 [[binary relation]]s, ''R''<sub>0</sub>, ''R''<sub>1</sub>, ..., ''R''<sub>''n''</sub> which satisfy:

*<math>R_{0} = \{(x,x) : x \in X\}</math>; it is called the [[identity relation]].
*Defining <math> R^* := \{(x,y) : (y,x) \in R\}</math>, if ''R'' in ''S'', then ''R*'' in ''S''.
*If <math>(x,y) \in R_{k}</math>, the number of <math>z \in X</math> such that <math>(x,z) \in R_{i}</math> and <math>(z,y) \in R_{j}</math> is a constant <math>p^k_{ij}</math> depending on <math>i</math>, <math>j</math>, <math>k</math> but not on the particular choice of <math>x</math> and <math>y</math>.

An association scheme is ''commutative'' if <math>p_{ij}^k = p_{ji}^k</math> for all <math>i</math>, <math>j</math> and <math>k</math>. Most authors assume this property.
Note, however, that while the notion of an association scheme generalizes the notion of a group, the notion of a commutative association scheme only generalizes the notion of a [[commutative group]].

A ''symmetric'' association scheme is one in which each <math>R_i</math> is a [[symmetric relation]]. That is:

* if (''x'', ''y'')  ∈ ''R''<sub>''i''</sub>, then (''y'', ''x'') ∈ ''R''<sub>''i''</sub>. (Or equivalently, ''R''* = ''R''.)
Every symmetric association scheme is commutative.

Two points ''x'' and ''y'' are called ''i''&thinsp;th associates if <math>(x,y) \in R_i</math>. The definition states that if ''x'' and ''y'' are ''i''&thinsp;th associates then so are ''y'' and ''x''. Every pair of points are ''i''&thinsp;th associates for exactly one <math>i</math>. Each point is its own zeroth associate while distinct points are never zeroth associates. If ''x'' and ''y'' are ''k''&thinsp;th associates then the number of points <math>z</math> which are both ''i''&thinsp;th associates of <math>x</math> and ''j''&thinsp;th associates of <math>y</math> is a constant <math>p^k_{ij}</math>.

=== Graph interpretation and adjacency matrices ===

A symmetric association scheme can be visualized as a [[complete graph]] with labeled edges. The graph has <math>v</math> vertices, one for each point of <math>X</math>, and the edge joining vertices <math>x</math> and <math>y</math> is labeled <math>i</math> if <math>x</math> and <math>y</math> are <math>i</math>&thinsp;th associates. Each edge has a unique label, and the number of triangles with a fixed base labeled <math>k</math> having the other edges labeled <math>i</math> and <math>j</math> is a constant <math>p^k_{ij}</math>, depending on <math>i,j,k</math> but not on the choice of the base. In particular, each vertex is incident with exactly <math>p^0_{ii}=v_{i}</math> edges labeled <math>i</math>; <math>v_{i}</math> is the [[Adjacency relation|valency]] of the [[Relation (mathematics)|relation]] <math>R_{i}</math>. There are also loops labeled <math>0</math> at each vertex <math>x</math>, corresponding to <math>R_{0}</math>.

The [[Relation (mathematics)|relations]] are described by their [[adjacency matrix|adjacency matrices]]. <math>A_i</math> is the adjacency matrix of <math>R_{i}</math> for <math>i=0,\ldots,n</math> and is a ''v'' × ''v'' [[Matrix (mathematics)|matrix]] with rows and columns labeled by the points of <math>X</math>.
:<math>\left( A_i \right)_{x,y} = \begin{cases}
1, & \mbox{if } (x,y) \in R_{i},\\ 
0, & \mbox{otherwise.}  \end{cases}\qquad (1)</math>

The definition of a symmetric association scheme is equivalent to saying that the <math>A_i</math> are ''v'' × ''v'' [[(0,1)-matrix|(0,1)-matrices]] which satisfy
:I. <math>A_i</math> is symmetric,
:II. <math>\sum_{i=0}^n A_i = J</math> (the all-ones matrix),
:III. <math>A_0 = I</math>,
:IV. <math>A_i A_j = \sum_{k=0}^n p^k_{ij}A_k = A_j A_i, i,j=0,\ldots,n</math>.

The (''x'', ''y'')-th entry of the left side of (IV) is the number of paths of length two between ''x'' and ''y'' with labels ''i'' and ''j'' in the graph. Note that the rows and columns of <math>A_{i}</math> contain <math>v_{i}</math> <math>1</math>'s:

:<math>A_{i} J=J A_{i}=v_{i} J. \qquad(2)</math>

=== Terminology ===
*The numbers <math>p_{ij}^k</math> are called the ''parameters'' of the scheme. They are also referred to as the ''structural constants''.

== History ==
The term ''association scheme'' is due to {{harv|Bose|Shimamoto|1952}} but the concept is already inherent in {{harv|Bose|Nair|1939}}.<ref>{{harvnb|Dembowski|1968|loc=pg. 281, footnote 1}}</ref> These authors were studying what statisticians have called ''partially balanced incomplete block designs'' (PBIBDs). The subject became an object of algebraic interest with the publication of {{harv|Bose|Mesner|1959}} and the introduction of the [[Bose–Mesner algebra]]. The most important contribution to the theory was the thesis of P. Delsarte {{harv|Delsarte|1973}} who recognized and fully used the connections with coding theory and design theory.<ref>{{harvnb|Bannai|Ito|1984|loc=pg. vii}}</ref> Generalizations have been studied by D. G. Higman (coherent configurations) and [[Boris Weisfeiler|B. Weisfeiler]] ([[distance regular graph]]s).

== Basic facts ==

*<math>p_{00}^0 = 1</math>, i.e., if <math>(x,y) \in R_0</math> then <math>x = y</math> and the only <math>z</math> such that <math>(x,z) \in R_0</math> is <math>z=x</math>.
*<math>\sum_{i=0}^{k} p_{ii}^0 = |X|</math>; this is because the <math>R_i</math> partition <math>X</math>.

== The Bose–Mesner algebra ==

The [[Adjacency matrix|adjacency matrices]] <math>A_i</math> of the [[Graph (discrete mathematics)|graphs]] <math>\left(X,R_{i}\right)</math> generate a [[commutative algebra (structure)|commutative]] and [[associative algebra]] <math>\mathcal{A}</math> (over the [[real number|real]] or [[complex number]]s) both for the [[matrix product]] and the [[Hadamard product (matrices)|pointwise product]]. This associative, commutative algebra is called the [[Bose–Mesner algebra]] of the association scheme.

Since the matrices in <math>\mathcal{A}</math> are [[symmetric matrix|symmetric]] and [[commuting matrices|commute]] with each other, they can be [[Diagonal matrix|diagonalized]] simultaneously. Therefore, <math>\mathcal{A}</math> is [[Semisimple operator|semi-simple]] and has a unique basis of primitive [[idempotent]]s <math>J_{0},\ldots,J_{n}</math>.

There is another algebra of <math>(n+1)\times(n+1)</math> matrices which is [[isomorphic]] to <math>\mathcal{A}</math>, and is often easier to work with.

== Examples ==

*The [[Johnson scheme]], denoted by ''J''(''v'', ''k''), is defined as follows.  Let ''S'' be a set with ''v'' elements.  The points of the scheme ''J''(''v'', ''k'') are the <math>{v \choose k}</math> [[subset]]s of S with ''k'' elements.  Two ''k''-element subsets ''A'', ''B'' of ''S'' are ''i''&thinsp;th associates when their intersection has size ''k''&nbsp;−&nbsp;''i''.
*The [[Hamming scheme]], denoted by ''H''(''n'', ''q''), is defined as follows.  The points of ''H''(''n'', ''q'') are the ''q<sup>n</sup>'' ordered ''n''-[[tuple]]s over a set of size ''q''.  Two ''n''-tuples ''x'', ''y'' are said to be ''i''&thinsp;th associates if they disagree in exactly ''i'' coordinates.  E.g., if ''x'' = (1,0,1,1), ''y'' = (1,1,1,1), ''z'' = (0,0,1,1), then ''x'' and ''y'' are 1st associates, ''x'' and ''z'' are 1st associates and ''y'' and ''z'' are 2nd associates in ''H''(4,2).
*A [[distance-regular graph]], ''G'', forms an association scheme by defining two vertices to be ''i''&thinsp;th associates if their distance is ''i''.
*A [[finite group]] ''G'' yields an association scheme on <math>X=G</math>, with a class ''R''<sub>''g''</sub> for each group element, as follows: for each <math>g \in G</math> let <math>R_g = \{(x,y) \mid x=g*y\}</math> where <math>*</math> is the group [[operation (mathematics)|operation]].  The class of the [[identity element|group identity]] is ''R''<sub>0</sub>.  This association scheme is commutative [[if and only if]] ''G'' is [[Abelian group|abelian]].
*A specific 3-class association scheme:<ref>{{harvnb|Street|Street|1987|loc=pg. 238}}</ref>
:Let ''A''(3) be the following association scheme with three associate classes on the set ''X'' = {1,2,3,4,5,6}. The (''i'', ''j''&hairsp;) entry is ''s'' if elements ''i'' and ''j'' are in relation ''R''<sub>''s''</sub>.
{| class="wikitable" style="margin:1em auto;"
! &nbsp;!! 1!! 2!! 3!! 4!! 5!! 6
|-
| '''1''' || <span style="color:white; background:blue"> &nbsp;0&nbsp;</span> || <span style="color:white; background:red"> &nbsp;1&nbsp; </span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:lime"> &nbsp;2&nbsp; </span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span>
|-
|'''2'''|| <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:blue"> &nbsp;0&nbsp; </span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:lime"> &nbsp;2&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span>
|-
| '''3''' || <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:blue"> &nbsp;0&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:lime"> &nbsp;2&nbsp;</span>
|-
| '''4''' || <span style="color:white; background:lime"> &nbsp;2&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:blue"> &nbsp;0&nbsp;</span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span>
|-
| '''5''' || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:lime"> &nbsp;2&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:blue">&nbsp;0&nbsp;</span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span>
|-
| '''6''' || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp; </span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:lime"> &nbsp;2&nbsp;</span> || <span style="color:white; background:red"> &nbsp;1&nbsp; </span> || <span style="color:white; background:red"> &nbsp;1&nbsp; </span> || <span style="color:white; background:blue">&nbsp;0&nbsp;</span>
|}

== Coding theory ==

The [[Hamming scheme]] and the [[Johnson scheme]] are of major significance in classical [[coding theory]].

In [[coding theory]], association scheme theory is mainly concerned with the [[Hamming distance|distance]] of a [[code]].  The [[linear programming]] method produces upper bounds for the size of a [[code]] with given minimum [[Hamming distance|distance]], and lower bounds for the size of a [[T-design|design]] with a given strength. The most specific results are obtained in the case where the underlying association scheme satisfies certain [[polynomial]] properties; this leads one into the realm of [[orthogonal polynomials]]. In particular, some universal bounds are derived for [[code]]s and [[T-design|designs]] in polynomial-type association schemes.

In classical [[coding theory]], dealing with [[code]]s in a [[Hamming scheme]], the MacWilliams transform involves a family of orthogonal polynomials known as the [[Krawtchouk polynomials]]. These polynomials give the [[eigenvalue]]s of the distance relation matrices of the [[Hamming scheme]].

== See also ==
<!-- * [[Analysis of variance]] (Statistics) -->
* [[Block design]]
* [[Bose–Mesner algebra]]
* [[Combinatorial design]]

== Notes ==
{{Reflist}}

== References ==

* {{citation| first=Rosemary A.| last=Bailey | author-link=Rosemary A. Bailey | url=http://www.maths.qmul.ac.uk/~rab/Asbook | title=Association Schemes: Designed Experiments, Algebra and Combinatorics|publisher=Cambridge University Press|year=2004|isbn=978-0-521-82446-0| mr=2047311}}. (Chapters from preliminary draft are [http://www.maths.qmw.ac.uk/~rab available on-line].)
* {{citation| last1=Bannai | first1=Eiichi | last2=Ito | first2=Tatsuro | title=Algebraic combinatorics I: Association schemes |  publisher=Benjamin/Cummings | location=Menlo Park, CA | year=1984 | isbn=0-8053-0490-8 | mr=0882540 }}
* {{citation| last1=Bose|first1=R. C.| author-link1=R. C. Bose| last2=Mesner|first2=D. M.|year=1959|title=On linear associative algebras corresponding to association schemes of partially balanced designs|journal=[[Annals of Mathematical Statistics]]|volume=30|issue=1|pages=21–38| url=http://projecteuclid.org/euclid.aoms/1177706356 | doi=10.1214/aoms/1177706356 | mr = 102157 | jstor = 2237117|doi-access=free}}
* {{citation| last1=Bose|first1=R.&nbsp;C.|author-link1=R. C. Bose|first2= K.&nbsp;R.|last2= Nair|title= Partially balanced incomplete block designs|journal= [[Sankhya (journal)|Sankhyā]]|volume= 4|year=1939|issue=3 |pages= 337–372 |jstor=40383923}}
* {{citation|last1=Bose|first1=R.&nbsp;C.|last2=Shimamoto|first2=T.|title=Classification and analysis of partially balanced incomplete block designs with two associate classes|journal=Journal of the American Statistical Association|year=1952|volume=47|issue=258|pages=151–184|author-link=R. C. Bose|doi=10.1080/01621459.1952.10501161}}
* {{citation |first=P. |last=Camion |chapter=18. Codes and Association Schemes: Basic Properties of Association Schemes Relevant to Coding |editor-first=V.S. |editor-last=Pless |editor2-first=W.C. |editor2-last=Huffman |editor3-first=R.A. |editor3-last=Brualdi  |title=Handbook of Coding Theory |publisher=Elsevier |volume=1 |year=1998 |isbn=978-0-444-50088-5 |pages=1441– }}
* {{citation|last=Delsarte |first= P.|year=1973 |title= An Algebraic Approach to the Association Schemes of Coding Theory|journal=  Philips Research Reports |issue=Supplement No. 10 |oclc=641852316 }}
* {{cite journal | last1 = Delsarte | first1 = P. | last2 = Levenshtein | first2 = V. I. | year = 1998 | title = Association schemes and coding theory | journal = IEEE Transactions on Information Theory | volume = 44 | issue = 6| pages = 2477–2504 | doi=10.1109/18.720545}}
* {{citation |first1=P. |last1=Dembowski |title=Finite Geometries |url=https://books.google.com/books?id=NJy_iTT_wGMC |year=1968 |publisher=Springer |isbn=978-3-540-61786-0}}
* {{citation|first=C. D.| last=Godsil|author-link = Chris Godsil|title=Algebraic Combinatorics|publisher=Chapman and Hall|year=1993|location=New York|isbn=0-412-04131-6 | mr=1220704 }}
* {{citation |first1=F.J. |last1=MacWilliams |first2=N.J.A. |last2=Sloane |title=The Theory of Error Correcting Codes |url=https://books.google.com/books?id=nv6WCJgcjxcC&pg=PP1 |year=1977 |publisher=Elsevier |isbn=978-0-444-85010-2 |series=North-Holland Mathematical Library |volume=16}}
*{{citation
|last1=Street|first1=Anne Penfold|author1-link=Anne Penfold Street  |last2=Street|first2= Deborah J.|author2-link= Deborah Street
 |title=Combinatorics of Experimental Design|title-link= Combinatorics of Experimental Design
|publisher=Oxford U. P. [Clarendon]
|year=1987
|isbn=0-19-853256-3
}}

* {{citation |last1=van Lint |first1=J.H. |last2=Wilson |first2=R.M. |year=1992 |title=A Course in Combinatorics |publisher=Cambridge University Press |isbn=0-521-00601-5}}
* {{citation|doi=10.1090/S0273-0979-05-01077-3|url=https://www.ams.org/bull/2006-43-02/S0273-0979-05-01077-3/S0273-0979-05-01077-3.pdf|title=''Association Schemes: Designed Experiments, Algebra and Combinatorics'' by Rosemary A. Bailey, Review|first= Paul-Hermann|last=Zieschang|journal=Bulletin of the American Mathematical Society|volume=43|year=2005a|issue=2|pages=249–253|doi-access=free}}
* {{citation| last=Zieschang | first=Paul-Hermann |  title=Theory of association schemes |  publisher=Springer |  year=2005b | isbn=3-540-26136-2 }}
* {{citation | last1=Zieschang | first1=Paul-Hermann | title=The exchange condition for association schemes | doi=10.1007/BF02777367 | doi-access=free | mr=2214129 | year=2006 | journal=[[Israel Journal of Mathematics]] | volume=151 | issue=3 | pages=357–380| s2cid=120009352 }}

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[[Category:Design of experiments]]
[[Category:Analysis of variance]]
[[Category:Algebraic combinatorics]]
[[Category:Representation theory]]