14 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bose–Mesner algebra | 2/2 | https://en.wikipedia.org/wiki/Bose–Mesner_algebra | reference | science, encyclopedia | 2026-05-05T09:49:22.162239+00:00 | kb-cron |
∑
i
=
0
n
J
i
=
I
.
(
5
)
{\displaystyle \sum _{i=0}^{n}J_{i}=I.\qquad (5)}
The Bose–Mesner algebra has two distinguished bases: the basis consisting of the adjacency matrices
D
i
{\displaystyle D_{i}}
, and the basis consisting of the irreducible idempotent matrices
J
k
{\displaystyle J_{k}}
. By definition, there exist well-defined complex numbers such that
D
i
=
∑
k
=
0
n
p
i
(
k
)
J
k
,
(
6
)
{\displaystyle D_{i}=\sum _{k=0}^{n}p_{i}(k)J_{k},\qquad (6)}
and
|
X
|
J
k
=
∑
i
=
0
n
q
k
(
i
)
D
i
.
(
7
)
{\displaystyle |X|J_{k}=\sum _{i=0}^{n}q_{k}\left(i\right)D_{i}.\qquad (7)}
The p-numbers
p
i
(
k
)
{\displaystyle p_{i}(k)}
, and the q-numbers
q
k
(
i
)
{\displaystyle q_{k}(i)}
, play a prominent role in the theory. They satisfy well-defined orthogonality relations. The p-numbers are the eigenvalues of the adjacency matrix
D
i
{\displaystyle D_{i}}
.
== Theorem == The eigenvalues of
p
i
(
k
)
{\displaystyle p_{i}(k)}
and
q
k
(
i
)
{\displaystyle q_{k}(i)}
, satisfy the orthogonality conditions:
∑
k
=
0
n
μ
i
p
i
(
k
)
p
ℓ
(
k
)
=
v
v
i
δ
i
ℓ
,
(
8
)
{\displaystyle \sum _{k=0}^{n}\mu _{i}p_{i}(k)p_{\ell }(k)=vv_{i}\delta _{i\ell },\quad (8)}
∑
k
=
0
n
μ
i
q
k
(
i
)
q
ℓ
(
i
)
=
v
μ
k
δ
k
ℓ
.
(
9
)
{\displaystyle \sum _{k=0}^{n}\mu _{i}q_{k}(i)q_{\ell }(i)=v\mu _{k}\delta _{k\ell }.\quad (9)}
Also
μ
j
p
i
(
j
)
=
v
i
q
j
(
i
)
,
i
,
j
=
0
,
…
,
n
.
(
10
)
{\displaystyle \mu _{j}p_{i}(j)=v_{i}q_{j}(i),\quad i,j=0,\ldots ,n.\quad (10)}
In matrix notation, these are
P
T
Δ
μ
P
=
v
Δ
v
,
(
11
)
{\displaystyle P^{T}\Delta _{\mu }P=v\Delta _{v},\quad (11)}
Q
T
Δ
v
Q
=
v
Δ
μ
,
(
12
)
{\displaystyle Q^{T}\Delta _{v}Q=v\Delta _{\mu },\quad (12)}
where
Δ
v
=
diag
{
v
0
,
v
1
,
…
,
v
n
}
,
Δ
μ
=
diag
{
μ
0
,
μ
1
,
…
,
μ
n
}
.
{\displaystyle \Delta _{v}=\operatorname {diag} \{v_{0},v_{1},\ldots ,v_{n}\},\qquad \Delta _{\mu }=\operatorname {diag} \{\mu _{0},\mu _{1},\ldots ,\mu _{n}\}.}
== Proof of theorem == The eigenvalues of
D
i
D
ℓ
{\displaystyle D_{i}D_{\ell }}
are
p
i
(
k
)
p
ℓ
(
k
)
{\displaystyle p_{i}(k)p_{\ell }(k)}
with multiplicities
μ
k
{\displaystyle \mu _{k}}
. This implies that
v
v
i
δ
i
ℓ
=
trace
D
i
D
ℓ
=
∑
k
=
0
n
μ
i
p
i
(
k
)
p
ℓ
(
k
)
,
(
13
)
{\displaystyle vv_{i}\delta _{i\ell }=\operatorname {trace} D_{i}D_{\ell }=\sum _{k=0}^{n}\mu _{i}p_{i}(k)p_{\ell }(k),\quad (13)}
which proves Equation
(
8
)
{\displaystyle \left(8\right)}
and Equation
(
11
)
{\displaystyle \left(11\right)}
,
Q
=
v
P
−
1
=
Δ
v
−
1
P
T
Δ
μ
,
(
14
)
{\displaystyle Q=vP^{-1}=\Delta _{v}^{-1}P^{T}\Delta _{\mu },\quad (14)}
which gives Equations
(
9
)
{\displaystyle (9)}
,
(
10
)
{\displaystyle (10)}
and
(
12
)
{\displaystyle (12)}
.
◻
{\displaystyle \Box }
There is an analogy between extensions of association schemes and extensions of finite fields. The cases we are most interested in are those where the extended schemes are defined on the
n
{\displaystyle n}
-th Cartesian power
X
=
F
n
{\displaystyle X={\mathcal {F}}^{n}}
of a set
F
{\displaystyle {\mathcal {F}}}
on which a basic association scheme
(
F
,
K
)
{\displaystyle \left({\mathcal {F}},K\right)}
is defined. A first association scheme defined on
X
=
F
n
{\displaystyle X={\mathcal {F}}^{n}}
is called the
n
{\displaystyle n}
-th Kronecker power
(
F
,
K
)
⊗
n
{\displaystyle \left({\mathcal {F}},K\right)_{\otimes }^{n}}
of
(
F
,
K
)
{\displaystyle \left({\mathcal {F}},K\right)}
. Next the extension is defined on the same set
X
=
F
n
{\displaystyle X={\mathcal {F}}^{n}}
by gathering classes of
(
F
,
K
)
⊗
n
{\displaystyle \left({\mathcal {F}},K\right)_{\otimes }^{n}}
. The Kronecker power corresponds to the polynomial ring
F
[
X
]
{\displaystyle F\left[X\right]}
first defined on a field
F
{\displaystyle \mathbb {F} }
, while the extension scheme corresponds to the extension field obtained as a quotient. An example of such an extended scheme is the Hamming scheme. Association schemes may be merged, but merging them leads to non-symmetric association schemes, whereas all usual codes are subgroups in symmetric Abelian schemes.
== See also == Association scheme
== Notes ==
== References == Bailey, Rosemary A. (2004), Association schemes: Designed experiments, algebra and combinatorics, Cambridge Studies in Advanced Mathematics, vol. 84, Cambridge University Press, p. 387, ISBN 978-0-521-82446-0, MR 2047311 Bannai, Eiichi; Ito, Tatsuro (1984), Algebraic combinatorics I: Association schemes, Menlo Park, CA: The Benjamin/Cummings Publishing Co., Inc., pp. xxiv+425, ISBN 0-8053-0490-8, MR 0882540 Bannai, Etsuko (2001), "Bose–Mesner algebras associated with four-weight spin models", Graphs and Combinatorics, 17 (4): 589–598, doi:10.1007/PL00007251, S2CID 41255028 Bose, R. C.; Mesner, D. M. (1959), "On linear associative algebras corresponding to association schemes of partially balanced designs", Annals of Mathematical Statistics, 30 (1): 21–38, doi:10.1214/aoms/1177706356, JSTOR 2237117, MR 0102157 Cameron, P. J.; van Lint, J. H. (1991), Designs, Graphs, Codes and their Links, Cambridge: Cambridge University Press, ISBN 0-521-42385-6 Camion, P. (1998), "Codes and association schemes: Basic properties of association schemes relevant to coding", in Pless, V. S.; Huffman, W. C. (eds.), Handbook of coding theory, The Netherlands: Elsevier Delsarte, P.; Levenshtein, V. I. (1998), "Association schemes and coding theory", IEEE Transactions on Information Theory, 44 (6): 2477–2504, doi:10.1109/18.720545 MacWilliams, F. J.; Sloane, N. J. A. (1978), The theory of error-correcting codes, New York: Elsevier Nomura, K. (1997), "An algebra associated with a spin model", Journal of Algebraic Combinatorics, 6 (1): 53–58, doi:10.1023/A:1008644201287