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Journal of Algebraic Combinatorics 7 (1998), 5–15
c 1998 Kluwer Academic Publishers. Manufactured in The Netherlands.
°
A Note on Thin P-Polynomial and Dual-Thin
Q-Polynomial Symmetric Association Schemes
GARTH A. DICKIE
dickie@win.tue.nl
University of Technology, Discrete Mathematics HG 9.53, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
PAUL M. TERWILLIGER
terwilli@math.wisc.edu
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison WI 53706
Received December 18, 1995
Abstract. Let Y denote a d-class symmetric association scheme, with d ≥ 3. We show the following: If Y
admits a P-polynomial structure with intersection numbers pihj and Y is 1-thin with respect to at least one vertex, then
1
i
= 0 ⇒ p1i
=0
p11
1 ≤ i ≤ d − 1.
If Y admits a Q-polynomial structure with Krein parameters qihj , and Y is dual 1-thin with respect to at least one
vertex, then
1
i
q11
= 0 ⇒ q1i
=0
1 ≤ i ≤ d − 1.
Keywords: Association scheme, distance-regular graph, intersection number, Q-polynomial
1.
Introduction
Let Y denote a d-class symmetric association scheme, with d ≥ 3. It is well-known that if
Y admits a P-polynomial structure with intersection numbers pihj , then
1
i
p11
6= 0 ⇒ p1i
6= 0
1≤i ≤d −1
(1)
[1, Theorem 5.5.1]. The first author shows in [3] that if Y admits a Q-polynomial structure
with Krein parameters qihj , then
1
q11
6= 0 ⇒ q1ii 6= 0
1 ≤ i ≤ d − 1.
(2)
In the present paper we show the following: If Y admits a P-polynomial structure with
intersection numbers pihj , and Y is 1-thin with respect to at least one vertex, then
1
i
= 0 ⇒ p1i
=0
p11
1 ≤ i ≤ d − 1.
(3)
If Y admits a Q-polynomial structure with Krein parameters qihj , and Y is dual 1-thin
with respect to at least one vertex, then
1
= 0 ⇒ q1ii = 0
q11
1 ≤ i ≤ d − 1.
(4)
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The 1-thin and dual 1-thin conditions are defined in Section 1.4. Our main results are in
Theorems 2.1 and 2.2.
In the following sections we introduce notation and recall basic results, following [1,
Section 2.1] and [4, Section 3].
1.1.
Symmetric association schemes
By a d-class symmetric association scheme we mean a pair Y = (X, {Ri }0≤i≤d ), where X
is a non-empty finite set, and where
(i)
(ii)
(iii)
(iv)
{Ri }0≤i≤d is a partition of X × X ;
R0 = {x x | x ∈ X };
Ri = Rit for 0 ≤ i ≤ d, where Rit = {yx | x y ∈ Ri };
there exist integers pihj such that for all integers h with 0 ≤ h ≤ d and all vertices
x, y ∈ X with x y ∈ Rh ,
pihj = |{z ∈ X | x z ∈ Ri , yz ∈ R j }|
0 ≤ i, j ≤ d.
(5)
We refer to X as the vertex set of Y , and refer to the integers pihj as the intersection numbers
of Y . Abbreviate ki = pii0 , and observe ki is non-zero for 0 ≤ i ≤ d.
1.2.
The Bose-Mesner algebra
Let Y = (X, {Ri }0≤i≤d ) denote a symmetric association scheme. Let Mat X (R) denote the
algebra of matrices over R with rows and columns indexed by X . The associate matrices
for Y are the matrices A0 , . . . , Ad ∈ Mat X (R) defined by
(
1 if x y ∈ Ri ,
(Ai )x y =
x, y ∈ X.
(6)
0 otherwise
From (i)–(iv) above we obtain
A0 + · · · + Ad = J,
Ai ◦ A j = δi j Ai 0 ≤ i, j ≤ d,
A0 = I,
Ai = Ait 0 ≤ i ≤ d,
d
X
Ai A j =
pihj Ah 0 ≤ i, j ≤ d,
(7)
(8)
(9)
(10)
(11)
h=0
where J is the all-1s matrix and ◦ denotes the entry-wise matrix product.
By the Bose-Mesner algebra of Y we mean the subalgebra M of Mat X (R) generated by
the associate matrices A0 , . . . , Ad . Observe by (8) and (11) that the associate matrices form
a basis for M. In particular, M is symmetric and closed under ◦.
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The algebra M has a second basis E 0 , . . . , E d such that
E 0 + · · · + E d = I,
E i E j = δi j E i 0 ≤ i, j ≤ d,
1
J,
E0 =
|X |
E i = E it 0 ≤ i ≤ d,
(12)
(13)
(14)
(15)
[1, Theorem 2.6.1]. We refer to E 0 , . . . , E d as the primitive idempotents of Y . Since M is
closed under ◦, there exist real numbers qihj satisfying
Ei ◦ E j =
d
1 X
q h Eh
|X | h=0 i j
0 ≤ i, j ≤ d.
(16)
The numbers qihj are the Krein parameters for Y . Abbreviate ki∗ = qii0 for 0 ≤ i ≤ d.
By (8), (9), and the fact that A0 , . . . , Ad is a basis for M, the primitive idempotents have
constant diagonal; in fact
(E i )x x =
ki∗
|X |
0 ≤ i ≤ d, x ∈ X
(17)
and ki∗ 6= 0 [1, p. 45]. We apply (17) in the proof of Lemma 4.1.
1.3.
The dual Bose-Mesner algebra
Let Y denote a d-class symmetric association scheme with vertex set X , associate matrices
A0 , . . . , Ad , primitive idempotents E 0 , . . . , E d , and Bose-Mesner algebra M. Fix a vertex
x ∈ X.
For each integer i with 0 ≤ i ≤ d let Ai∗ = Ai∗ (x) denote the diagonal matrix in Mat X (R)
defined by
(Ai∗ ) yy = |X |(E i )x y
y ∈ X.
(18)
We refer to A∗0 , . . . , A∗d as the dual associate matrices for Y with respect to x. Let
M = M ∗ (x) denote the subalgebra of Mat X (R) generated by the dual associate matrices.
We refer to M ∗ as the dual Bose-Mesner algebra for Y with respect to x. From (16) we
obtain
∗
Ai∗ A∗j =
d
X
qihj A∗h
0 ≤ i, j ≤ d.
h=0
In particular, the dual associate matrices form a basis for M ∗ .
(19)
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For each integer i with 0 ≤ i ≤ d let E i∗ = E i∗ (x) denote the diagonal matrix in Mat X (R)
defined by
(E i∗ ) yy = (Ai )x y
y ∈ X.
(20)
From (7), (8) we obtain
E 0∗ + · · · + E d∗ = I,
E i∗ E ∗j = δi j E i∗
0 ≤ i, j ≤ d.
(21)
(22)
We refer to E 0∗ , . . . , E d∗ as the dual idempotents for Y with respect to x. Note that the
dual idempotents form a second basis for M ∗ .
1.4.
The thin and dual-thin conditions
Let Y denote a d-class symmetric association scheme with vertex set X . Fix a vertex x ∈ X ,
and write M ∗ = M ∗ (x).
Let T = T (x) denote the subalgebra of Mat X (R) generated by M and M ∗ . We refer to T
as the subconstituent algebra for Y with respect to x. By a T -module we mean a subspace
of the standard module V = R X which is closed under multiplication by T . A T -module
is said to be irreducible if it properly contains no T -modules other than 0. Recall that T is
semi-simple, so that V may be decomposed as a direct sum of irreducible T -modules [4,
Lemma 3.4].
An irreducible T -module W is said to be thin if
dim E i∗ W ≤ 1
0 ≤ i ≤ d,
(23)
0 ≤ i ≤ d.
(24)
and dual thin if
dim E i W ≤ 1
We say Y is i-thin with respect to x if every irreducible T -module W with E i∗ W 6= 0
is thin. We say Y is dual i-thin with respect to x if every irreducible T -module W with
E i W 6= 0 is dual thin.
1.5.
P- and Q-polynomial structures
Let Y denote a d-class symmetric association scheme, with vertex set X , intersection
numbers pihj , and Krein parameters qihj . We say that an ordering A0 , . . . , Ad of the associate
matrices is a P-polynomial structure for Y whenever
pihj = 0
pihj
6= 0
if one of h, i, j is greater than the sum of the other two,
(25)
if one of h, i, j is equal to the sum of the other two
(26)
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for 0 ≤ h, i, j ≤ d. Recall that if A0 , . . . , Ad is a P-polynomial structure for Y , then A1
generates M [4, Lemma 3.8].
We say that an ordering E 0 , . . . , E d of the primitive idempotents is a Q-polynomial
structure for Y whenever
qihj = 0
qihj
6= 0
if one of h, i, j is greater than the sum of the other two,
(27)
if one of h, i, j is equal to the sum of the other two
(28)
for 0 ≤ h, i, j ≤ d. Recall that if E 0 , . . . , E d is a Q-polynomial structure for Y , then for
each x ∈ X the dual associate matrix A∗1 (x) generates M ∗ (x) [4, Lemma 3.11].
2.
Results
Our main results are the following:
Theorem 2.1 Let Y denote a d-class symmetric association scheme, with d ≥ 3. Suppose
A0 , . . . , Ad is a P-polynomial structure for Y with intersection numbers pihj , and suppose
Y is 1-thin with respect to at least one vertex. Then
1
i
p11
= 0 ⇒ p1i
=0
1 ≤ i ≤ d − 1.
(29)
We prove Theorem 2.1 in Section 3.
Theorem 2.2 Let Y denote a d-class symmetric association scheme, with d ≥ 3. Suppose
E 0 , . . . , E d is a Q-polynomial structure for Y with Krein parameters qihj , and suppose Y is
dual 1-thin with respect to at least one vertex. Then
1
q11
= 0 ⇒ q1ii = 0
1 ≤ i ≤ d − 1.
(30)
We prove Theorem 2.2 in Section 4.
3.
Proof of Theorem 2.1
Define a symmetric bilinear form on Mat X (R) (where X is any set) by
hB, Ci = tr(B t C)
B, C ∈ Mat X (R).
(31)
Observe that hB, Ci is just the sum of the entries of B ◦ C. In particular, the form is
positive definite.
Lemma 3.1 (Terwilliger [4]) Let Y = (X, {Ri }0≤i≤d ) denote a symmetric association
scheme with associate matrices A0 , . . . , Ad and intersection numbers pihj . Fix a vertex
x ∈ X , and write E i∗ = E i∗ (x) for 0 ≤ i ≤ d. Then:
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(i) for 0 ≤ h, h 0 , i, i 0 , j, j 0 ≤ d,
hE i∗ Ah E ∗j , E i∗0 Ah 0 E ∗j 0 i = δhh 0 δii 0 δ j j 0 kh pihj ;
(32)
(ii) for 0 ≤ h, i, j ≤ d,
E h∗ Ai E ∗j = 0 ⇔ pihj = 0.
(33)
Proof of (i): Observe
(E i∗ Ah E ∗j ) yz = (E i∗ ) yy (Ah ) yz (E ∗j )zz
= (Ai )x y (Ah ) yz (A j )x z ,
(34)
(35)
so that (E i∗ Ah E ∗j ) yz 6= 0 if and only if x y ∈ Ri , yz ∈ Rh , and x z ∈ R j . Since the relations
R0 , . . . , Rd are disjoint, the matrices E i∗ Ah E ∗j and E i∗0 Ah 0 E ∗j 0 have no non-zero entries in
common unless h = h 0 , i = i 0 , j = j 0 . In this case there are precisely kh pihj non-zero
entries, each equal to 1. The result follows.
Proof of (ii): Immediate from (i).
2
Let Y denote a d-class symmetric association scheme, with vertex set X . Suppose
A0 , . . . , Ad is a P-polynomial structure for Y , with intersection numbers pihj . Fix a vertex
x ∈ X , and write T = T (x), M ∗ = M ∗ (x), and E i∗ = E i∗ (x) for 0 ≤ i ≤ d.
There are three matrices in T which are of particular interest to us (their duals will be
used in Section 4). These are the lowering matrix L = L(x), the flat matrix F = F(x),
and the raising matrix R = R(x), defined by
L=
d
X
∗
E i−1
A1 E i∗ ,
(36)
E i∗ A1 E i∗ ,
(37)
∗
E i+1
A1 E i∗ .
(38)
i=1
F=
d
X
i=0
R=
d−1
X
i=0
It is easily shown using (25), (21), and (33) that
A1 = L + F + R.
(39)
Recall that A1 generates the Bose-Mesner algebra M, so that A1 and E 0∗ , . . . , E d∗ generate
T . In particular, L, F, R, and E 0∗ , . . . , E d∗ generate T by (39).
Lemma 3.2 Let Y denote a d-class symmetric association scheme, with vertex set X .
Suppose A0 , . . . , Ad is a P-polynomial structure for Y , with intersection numbers pihj . Fix
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a vertex x ∈ X , and write T = T (x), L = L(x), and E i∗ = E i∗ (x) for 0 ≤ i ≤ d. If Y is
1-thin with respect to x, then:
(i) for any irreducible T -module W with E 1∗ W 6= 0,
L E i∗ W = 0 ⇒ E i∗ W = 0
2 ≤ i ≤ d;
(40)
(ii) for w ∈ T E 1∗ V ,
L E i∗ w = 0 ⇒ E i∗ w = 0 2 ≤ i ≤ d;
(41)
(iii) for B ∈ T E 1∗ ,
L E i∗ B = 0 ⇒ E i∗ B = 0 2 ≤ i ≤ d.
(42)
Proof of (i): Let W be given. Fix an integer i with 2 ≤ i ≤ d, and suppose L E i∗ W = 0.
Let W 0 denote the subspace of W defined by
W 0 = E i∗ W + · · · + E d∗ W.
(43)
Observe by (36)–(38) and (13) that W 0 is closed under multiplication by L, F, R, and
T -module. Since E 1∗ W 0 = 0
is irreducible, we now have
E 0∗ , . . . , E d∗ . Since T is generated by these matrices, W 0 is a
and E 1∗ W 6= 0, W 0 is a proper submodule of W . Since W
W 0 = 0, and E i∗ W ⊆ W 0 is zero as desired.
Proof of (ii): Since V may be decomposed into a direct sum of irreducible T -modules, it
suffices to show that the result holds for w ∈ T E 1∗ W where W is an irreducible T -module.
Fix an integer i with 2 ≤ i ≤ d and an irreducible T -module W , and suppose w ∈ T E 1∗ W
has L E i∗ w = 0.
Suppose E i∗ w 6= 0. Observe E 1∗ W 6= 0, since 0 6= E i∗ w ∈ E i∗ T E 1∗ W . Since Y is 1-thin
with respect to x, W is thin and dim E i∗ W ≤ 1. In particular, E i∗ w ∈ E i∗ W spans E i∗ W , and
L E i∗ W = 0. By (i) we have E i∗ W = 0, and E i∗ w = 0 for a contradiction. Thus E i∗ w = 0
as desired.
Proof of (iii): Immediate from (ii).
2
Lemma 3.3 Let Y denote a d-class symmetric association scheme, with vertex set X .
Suppose A0 , . . . , Ad is a P-polynomial structure for Y , with intersection numbers pihj . Fix
a vertex x ∈ X , and write L = L(x) and E i∗ = E i∗ (x) for 0 ≤ i ≤ d. Then:
(i) for 1 ≤ i ≤ d − 1,
i
∗
E i−1
Ai E 1∗ ;
L E i∗ Ai+1 E 1∗ = p1,i+1
(44)
i−1
(ii) for 1 ≤ i ≤ d, if p1,i−1
= 0 then
i
∗
E i−1
Ai E 1∗ .
L E i∗ Ai E 1∗ = p1i
(45)
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Proof of (i): Let i be given. Observe by (22), (25), (33), (21), and (11) that
∗
L E i∗ Ai+1 E 1∗ = E i−1
AE i∗ Ai+1 E 1∗
Ã
!
d
X
∗
∗
E h Ai+1 E 1∗
= E i−1 A
(46)
(47)
h=0
∗
A Ai+1 E 1∗
= E i−1
Ã
!
d
X
∗
h
p1,i+1 Ah E 1∗
= E i−1
=
h=0
i
∗
E i−1
Ai E 1∗ ,
p1,i+1
(48)
(49)
(50)
as desired.
i−1
= 0. Observe as in (i) that
Proof of (ii): Let i be given, with p1,i−1
∗
AE i∗ Ai E 1∗
L E i∗ Ai E 1∗ = E i−1
Ã
!
d
X
∗
∗
= E i−1 A
E h Ai E 1∗
=
=
=
h=0
∗
A Ai E 1∗
E i−1
Ã
!
d
X
∗
h
p1i Ah
E i−1
h=0
i
∗
E i−1
Ai E 1∗ ,
p1i
(51)
(52)
(53)
E 1∗
as desired.
(54)
(55)
2
Proof of Theorem 2.1: Suppose Y is 1-thin with respect to x, and write L = L(x) and
i
1
= 0, and suppose for a contradiction that p1i
6= 0
E i∗ = E i∗ (x) for 0 ≤ i ≤ d. Suppose p11
i
6= 0. Then by Lemma 3.3,
for some i with 2 ≤ i ≤ d − 1. Fix i ≥ 2 minimal with p1i
¡ i ∗
¢
i
E i Ai+1 E 1∗ − p1,i+1
E i∗ Ai E 1∗ ,
0 = L p1i
(56)
and by Lemma 3.2(iii),
i
i
E i∗ Ai+1 E 1∗ − p1,i+1
E i∗ Ai E 1∗ .
0 = p1i
(57)
The summands in (57) are nonzero by (33) and orthogonal by (32), for a contradiction.
i
= 0 for 2 ≤ i ≤ d − 1, as desired.
2
Thus p1i
4.
Proof of Theorem 2.2
Our proof is based upon the following result:
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Lemma 4.1 (Cameron, Goethals, Seidel [2]) Let Y denote a d-class symmetric association scheme, with vertex set X , primitive idempotents E 0 , . . . , E d , and Krein parameters
qihj . Fix a vertex x ∈ X , and write Ai∗ = Ai∗ (x) for 0 ≤ i ≤ d. Then:
(i) for 0 ≤ h, h 0 , i, i 0 , j, j 0 ≤ d,
hE i A∗h E j , E i 0 A∗h 0 E j 0 i = δhh 0 δii 0 δ j j 0 kh∗ qihj ;
(58)
(ii) for 0 ≤ h, i, j ≤ d,
E h Ai∗ E j = 0 ⇔ qihj = 0.
(59)
Proof of (i): Recall tr(BC) = tr(C B), and observe by (15), (13), (18), (16), and (17) that
hE i A∗h E j , E i 0 A∗h 0 E j 0 i = tr(E j A∗h E i E i 0 A∗h 0 E j 0 )
= tr(E j 0 E j A∗h E i E i 0 A∗h 0 )
= δii 0 δ j j 0 tr(E j A∗h E i A∗h 0 )
X
(E j ) yz (A∗h )zz (E i )zy (A∗h 0 ) yy
= δii 0 δ j j 0
y,z∈X
= δii 0 δ j j 0 |X |2
X
(E j ) yz (E h )x z (E i )zy (E h 0 )x y
(60)
(61)
(62)
(63)
(64)
y,z∈X
= δii 0 δ j j 0 |X |2
X
((E i ◦ E j )E h ) yx (E h 0 )x y
(65)
y∈X
= δii 0 δ j j 0 |X |qihj
X
(E h ) yx (E h 0 )x y
(66)
y∈X
= δii 0 δ j j 0 |X |qihj (E h 0 E h )x x
(67)
= δhh 0 δii 0 δ j j 0 |X |qihj (E h )x x
(68)
=
δhh 0 δii 0 δ j j 0 kh∗ qihj ,
(69)
as desired.
Proof of (ii): Immediate from (i).
2
Let Y denote a d-class symmetric association scheme, with vertex set X . Suppose
E 0 , . . . , E d is a Q-polynomial structure for Y , with Krein parameters qihj . Fix a vertex
x ∈ X , and write T = T (x), M ∗ = M ∗ (x), and Ai∗ = Ai∗ (x) for 0 ≤ i ≤ d.
The dual lowering matrix L ∗ = L ∗ (x), the dual flat matrix F ∗ = F ∗ (x), and the dual
raising matrix R ∗ = R ∗ (x) are defined by
L∗ =
d
X
i=1
E i−1 A∗1 E i ,
(70)
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F∗ =
d
X
E i A∗1 E i ,
(71)
E i+1 A∗1 E i .
(72)
i=0
R∗ =
d−1
X
i=0
It is easily shown using (27), (12), and (59) that
A∗1 = L ∗ + F ∗ + R ∗ .
(73)
Recall that A∗1 generates the dual Bose-Mesner algebra M ∗ , so that A∗1 and E 0 , . . . , E d
generate T . In particular, L ∗ , F ∗ , R ∗ , and E 0∗ , . . . , E d∗ generate T by (73).
Lemma 4.2 Let Y denote a d-class symmetric association scheme, with vertex set X .
Suppose E 0 , . . . , E d is a Q-polynomial structure for Y , with Krein parameters qihj . Fix a
vertex x ∈ X , and write T = T (x), L ∗ = L ∗ (x), and Ai∗ = Ai∗ (x) for 0 ≤ i ≤ d. If Y is
dual 1-thin with respect to x, then:
(i) for any irreducible T -module W with E 1 W 6= 0,
L ∗ Ei W = 0 ⇒ Ei W = 0
2 ≤ i ≤ d;
(74)
(ii) for w ∈ T E 1 V ,
L ∗ Ei w = 0 ⇒ Ei w = 0
2 ≤ i ≤ d;
(75)
2 ≤ i ≤ d.
(76)
(iii) for B ∈ T E 1 ,
L ∗ Ei B = 0 ⇒ Ei B = 0
Proof: Similar to the proof of Lemma 3.2.
2
Lemma 4.3 Let Y denote a d-class symmetric association scheme, with vertex set X .
Suppose E 0 , . . . , E d is a Q-polynomial structure for Y , with Krein parameters qihj . Fix a
vertex x ∈ X , and write L ∗ = L ∗ (x) and Ai∗ = Ai∗ (x) for 0 ≤ i ≤ d. Then:
(i) for 1 ≤ i ≤ d,
∗
i
E 1 = q1,i+1
E i−1 Ai∗ E 1 ;
L ∗ E i Ai+1
(77)
i−1
= 0 then
(ii) for 1 ≤ i ≤ d, if q1,i−1
L ∗ E i Ai∗ E 1 = q1ii E i−1 Ai∗ E 1 .
(78)
Proof: Similar to the proof of Lemma 3.3.
2
Proof of Theorem 2.2: Similar to the proof of Theorem 2.1.
2
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References
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2. P. Cameron, J. Goethals, and J. Seidel, “The Krein condition, spherical designs, Norton algebras, and permutation groups,” Indag. Math. 40 (1978), 196–206.
3. G. Dickie. “A note on Q-polynomial association schemes,” J. Alg. Combin. Submitted.
4. P. Terwilliger. “The subconstituent algebra of an association scheme. I,” J. Alg. Combin. 1(4) (1992), 363–388.