Download The main theorem

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The key fact for dealing with two or more matrices in A is that A is commutative; that is, if M ∈ A and N ∈ A then MN = NM. This follows from Lemma 1.2.
The following rather technical lemma is the building block for the main result.
Lemma 2.4 Let U1 , . . . , Um be non-zero orthogonal subspaces of RΩ such that
RΩ = U1 ⊕ · · · ⊕ Um . For 1 6 i 6 m, let Qi be the orthogonal projector onto Ui .
Let M be a symmetric matrix which commutes with Qi for 1 6 i 6 m. Let the
eigenspaces of M be V1 , . . . , Vr . Let the non-zero subspaces among the intersections Ui ∩V j be W1 , . . . , Wt . Then
(i) RΩ = W1 ⊕ · · · ⊕Wt ;
(ii) if k 6= l then Wk ⊥ Wl ;
(iii) for 1 6 k 6 t, the space Wk is contained in an eigenspace of M;
(iv) for 1 6 k 6 t, the orthogonal projector onto Wk is a polynomial in Q1 , . . . ,
Qm and M.
Proof Part (iii) is immediate from the definition of the Wk . So is part (ii), because
Ui1 ⊥ Ui2 if i1 6= i2 and V j1 ⊥ V j2 if j1 6= j2 so the two spaces Ui1 ∩V j1 and Ui2 ∩V j2
are either orthogonal or equal.
For 1 6 j 6 r, let Pj be the projector onto V j . Since Pj is a polynomial in M,
it commutes with Qi , so Ui is geometrically orthogonal to V j . By Lemma 2.2, the
orthogonal projector onto Ui ∩V j is Qi Pj , which is a polynomial in Qi and M. This
proves (iv).
Finally, Qi Pj = O precisely when Ui ∩ Pj = {0}, so the sum of the orthogonal
projectors onto the spaces Wk is the sum of the non-zero products Qi Pj , which is
equal to the sum of all the products Qi Pj . But
!
!
∑ ∑ Qi P j = ∑ Qi
i
j
i
∑ Pj
= I 2 = I,
j
so Lemma 2.3 shows that RΩ = W1 ⊕ · · · ⊕Wt , proving (i).
Corollary 2.5 If M and N are commuting symmetric matrices in RΩ×Ω then RΩ
is the direct sum of the non-zero intersections of the eigenspaces of M and N.
Moreover, these spaces are mutually orthogonal and their orthogonal projectors
are polynomials in M and N.
Proof Apply Lemma 2.4 to M and the eigenspaces of N.
If W is contained in an eigenspace of a matrix M, I shall call it a sub-eigenspace
of M.
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Theorem 2.6 Let A be the Bose-Mesner algebra of an association scheme on Ω
with s associate classes and adjacency matrices A0 , A1 , . . . , As . Then RΩ has s +1
mutually orthogonal subspaces W0 , W1 , . . . , Ws , called strata, with orthogonal
projectors S0 , S1 , . . . , Ss such that
(i) RΩ = W0 ⊕W1 ⊕ · · · ⊕Ws ;
(ii) each of W0 , W1 , . . . , Ws is a sub-eigenspace of every matrix in A ;
(iii) for i = 0, 1, . . . , s, the adjacency matrix Ai is a linear combination of S0 , S1 ,
. . . , Ss ;
(iv) for e = 0, 1, . . . , s, the stratum projector Se is a linear combination of A0 ,
A1 , . . . , As .
Proof The adjacency matrices A1 , . . . , As commute and are symmetric, so s − 1
applications of Lemma 2.4, starting with the eigenspaces of A1 , give spaces W0 ,
. . . , Wr as the non-zero intersections of the eigenspaces of A1 , . . . , As , where r is
as yet unknown. These spaces We are mutually orthogonal and satisfy (i). Since
A0 = I and every matrix in A is a linear combination of A0 , A1 , . . . , As , the spaces
We clearly satisfy (ii). Each Se is a polynomial in A1 , . . . , As , hence in A , so (iv) is
satisfied. Let C(i, e) be the eigenvalue of Ai on We . Then Equation (2.2) shows
that
r
Ai =
∑ C(i, e)Se
e=0
and so (iii) is satisfied.
Finally, (iii) shows that S0 , . . . , Sr span A . Suppose that there are real scalars
λ0 , . . . , λr such that ∑e λe Se = O. Then for f = 0, . . . , r we have O = (∑e λe Se ) S f =
λ f S f so λ f = 0. Hence S0 , . . . , Sr are linearly independent, so they form a basis
for A . Thus r + 1 = dim A = s + 1 and r = s.
We now have two bases for A : the adjacency matrices and the stratum projectors. The former are useful for addition, because A j (α, β) = 0 if (α, β) ∈ Ci and
i 6= j. The stratum projectors make multiplication easy, because Se Se = Se and
Se S f = O if e 6= f .
Before calculating any eigenvalues, we
note that if Ais the adjacency matrix
of any subset C of Ω × Ω then Aχα = ∑ χβ : (β, α) ∈ C . In particular, Ai χα =
χCi (α) and Jχα = χΩ . Furthermore, if M is any matrix in RΩ×Ω then MχΩ =
∑ω∈Ω Mχω . If M has constant row-sum r then MχΩ = rχΩ .
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Example 2.2 Consider the
groups of size k and
n
1
A1 (α, β) =
0
n
1
A2 (α, β) =
0
group divisible association scheme GD(b, k) with b
if α and β are in the same group but α 6= β
otherwise
if α and β are in different groups
otherwise.
Consider the 1-dimensional space W0 spanned by χΩ . We have
A0 χΩ = IΩ χΩ =
χΩ
A1 χΩ = a1 χΩ = (k − 1)χΩ
A2 χΩ = a2 χΩ = (b − 1)kχΩ .
Thus W0 is a sub-eigenspace of every adjacency matrix. This does not prove that
W0 is a stratum, because there might be other vectors which are also eigenvectors
of the all the adjacency matrices with the same eigenvalues as W0 .
Let α and β be in the same group ∆. Then
A1 χα = χ∆ − χα
A2 χα = χΩ − χ∆
A1 χβ = χ∆ − χβ
A2 χβ = χΩ − χ∆
so A1 (χα − χβ ) = −(χα − χβ ) and A2 (χα − χβ ) = 0. Let Wwithin be the b(k − 1)dimensional “within groups” subspace spanned by all vectors of the form χα − χβ
with α and β in the same group. Then Wwithin is a sub-eigenspace of A0 , A1 and
A2 and the eigenvalues for A1 and A2 are different from those for W0 .
Since eigenspaces are mutually orthogonal, it is natural to look at the orthogonal complement of W0 +Wwithin . This is the (b−1)-dimensional “between groups”
subspace Wbetween spanned by vectors of the form χ∆ − χΓ where ∆ and Γ are different groups. Now
A1 χ∆ = A1
and
∑ χα = kχ∆ − ∑ χα = (k − 1)χ∆
α∈∆
A2 χ∆ = A2
α∈∆
∑ χα = k(χΩ − χ∆)
α∈∆
so A1 (χ∆ −χΓ ) = (k −1)(χ∆ −χΓ ) and A2 (χ∆ −χΓ ) = −k(χ∆ −χΓ ). Thus Wbetween
is a sub-eigenspace with different eigenvalues from Wwithin . Therefore the strata
are W0 , Wwithin and Wbetween .
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Lemma 2.7 If P ∈ A and P is idempotent then P = ∑e∈F Se for some subset F of
{0, . . . , s}.
Proof Let P = ∑se=0 λe Se . Then P2 = ∑se=0 λ2e Se , which is equal to P if and only
if λe ∈ {0, 1} for e = 0, . . . , s.
For this reason the stratum projectors are sometimes called minimal idempotents or primitive idempotents.
Lemma 2.8 The space W spanned by χΩ is always a stratum. Its projector is
|Ω|−1 JΩ .
Proof The orthogonal projector onto W is |Ω|−1 JΩ because
JΩ χΩ =
and
∑ JΩχω = |Ω| χΩ
ω∈Ω
JΩ (χα − χβ ) = 0.
This is an idempotent contained in A , so it is equal to ∑e∈F Se , for some subset F
of {0, . . . , s}, by Lemma 2.7. Then
1 = dimW = tr |Ω|−1 JΩ = ∑ tr Se = ∑ dimWe
e∈F
e∈F
so we must have |F | = 1 and W is itself a stratum.
Notation The 1-dimensional stratum spanned by χΩ is always called W0 .
Although there are the same number of strata as associate classes, there is
usually no natural bijection between them. When I want to emphasize this, I shall
use a set K to index the associate classes and a set E to index the strata. However,
there are some association schemes for which E and K are naturally the same but
for which W0 does not correspond to A0 . So the reader should interpret these two
subscripts ‘0’ as different sorts of zero.
I shall always write de for dimWe .