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Journal of Algebraic Combinatorics 7 (1998), 53–75
c 1998 Kluwer Academic Publishers. Manufactured in The Netherlands.
°
The Structure of Nonthin Irreducible
T-modules of Endpoint 1: Ladder Bases
and Classical Parameters
S. HOBART
Department of Mathematics, University of Wyoming, Laramie, WY 82071
T. ITO
Department of Computational Science, Kanazawa University, Kakuna-niachi, Kanazawa 920-11 Japan
Received June 29, 1995; Revised October 22, 1996
Abstract. Building on the work of Terwilliger, we find the structure of nonthin irreducible T -modules of endpoint
1 for P- and Q-polynomial association schemes with classical parameters. The isomorphism class of such a given
module is determined by the intersection numbers of the scheme and one additional parameter which must be an
eigenvalue for the first subconstituent graph. We show that these modules always have what we call a ladder basis,
and find the structure explicitly for the bilinear, Hermitean, and alternating forms schemes.
Keywords: Association scheme, Terwilliger algebra
1.
Introduction
The study of Terwilliger algebras for association schemes was begun by Terwilliger in
[4], where they are called subconstituent algebras. These noncommutative algebras are
generated by the Bose-Mesner algebra of the scheme, together with matrices containing
local information about the structure with respect to a fixed vertex. It is expected that these
algebras will contribute significantly to the classification of P- and Q-polynomial schemes.
The irreducible modules for thin P- and Q-polynomial schemes were thoroughly investigated by Terwilliger in [4]. Roughly speaking, he shows that such modules inherit the
P- and Q-polynomial property and have structures described by Askey–Wilson polynomials
related to those of the scheme. He also relates thinness to the combinatorial structure of the
scheme.
Little is known about nonthin irreducible modules, and their structures seem to be much
more complicated. For one particular family of schemes, the Doob schemes, all irreducible
modules were found by Tanabe [3]. However for the classical forms schemes (bilinear,
alternating, Hermitean, and quadratic forms), which for diameter ≥6 are the other known
examples of nonthin P- and Q-polynomial association schemes [4], the irreducible modules
have not yet been determined.
Some basic theory for the case of endpoint 1, is found in Terwilliger’s unpublished lecture
notes [5]. In particular, he shows that the isomorphism classes of modules are determined
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HOBART AND ITO
by the eigenvalues of the subgraph on the first subconstituent, and that they are only a
little larger than thin modules, in the sense that they intersect the distance i subspaces in
dimension 2 for 2 ≤ i ≤ d − 1 and dimension 1 for i = 1 and i = d.
In this paper, we consider only schemes with classical parameters; for these schemes,
we describe the nonthin irreducible modules of endpoint 1. We show that for classical
parameters, there exists a particularly nice basis for any nonthin irreducible module of
endpoint 1, which we call a ladder basis. Such bases were first shown to exist for modules
of the Doob schemes in [3]; part of the motivation for our work was a question of Terwilliger
as to whether such bases exist in general. In fact, we show something stronger: for classical
parameters, half of the elements of a ladder basis are multiples of elements of the basis
given in [5] (the Terwilliger basis). We also find the matrix for the action of the adjacency
matrix A1 on the module, with respect to both the Terwilliger and ladder bases.
All the known examples of P- and Q-polynomial schemes with diameter ≥6 which are
not thin have classical parameters. They are also self-dual, and so have what we may call
dual classical parameters for the Q-polynomial, or dual, structure. Since our methods are
algebraic rather than combinatorial, the theorems have dual versions (describing A∗1 instead
of A1 ) for schemes with dual classical parameters. However, we do not explore this further
in this paper.
Terwilliger algebras contain a lot of information about the scheme, possibly enough to reconstruct the scheme if it is P- and Q-polynomial. However, the determination of all nonthin
modules is extremely difficult in general. The advantage of investigating irreducible modules of endpoint 1 is that they are small enough to be manageable, but still can be expected
to reflect the nature of the local structure in the scheme. We hope that the investigation
of irreducible modules of small endpoint, together with combinatorial methods, will be
sufficient to finish the classification of P- and Q-polynomial schemes.
The organization of the paper is as follows. Section 2 gives definitions and a summary
of previous results from [4] and [5]. In Section 3, we define the term ladder basis, and give
necessary and sufficient conditions for existence. Section 4 shows that if the scheme has
classical parameters, any nonthin irreducible T -module of endpoint 1 for the scheme has a
ladder basis. In Section 5, we explicitly find the action of A1 on the module, with respect
to both the Terwilliger and ladder bases. The final section is devoted to examples: we find
the eigenvalues of the subgraph on the first subconstituent for the bilinear, alternating, and
Hermitean forms schemes, and use these to determine the nonthin irreducible endpoint 1
modules.
2.
Terwilliger algebras
In this section, we review the definition and basic results for Terwilliger algebras from [4]
and [5]. The books [1] and [2] are basic references for P- and Q-polynomial association
schemes.
2.1.
Definitions and notation
Let X = (X, {Ri }0≤i≤d ) be a commutative association scheme with d classes. As usual, let
Ai be the adjacency matrix for relation Ri , A the linear span of {A0 , A1 , . . . , Ad } over C,
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d
the set of primitive idempotents of A. E i is
that is, the Bose-Mesner algebra, and {E i }i=0
the projection onto the ith common eigenspace of the adjacency matrices.
Fix x ∈ X . Let 0i (x) be the ith subconstituent of X , so 0i (x) = {y ∈ X | (x, y) ∈ Ri }.
Define E i∗ = E i∗ (x) ∈ Mat X (C) to be the |X | × |X | diagonal matrix with (y, y) entry 1
if (x, y) ∈ Ri and 0 otherwise. The Terwilliger algebra T = T (x) is the subalgebra of
Mat X (C) generated by {A0 , . . . , Ad , E 0∗ , . . . , E d∗ }.
Let V = C|X | be the unitary space over C with an orthonormal basis which we identify
with X , and inner product h , i. V is a module for T , called the standard module. For
0 ≤ i ≤ d, V has a subspace Vi∗ = Vi∗ (x) with basis 0i (x); E i∗ is the orthogonal projection
onto Vi∗ .
T is semi-simple, so V decomposes into an orthogonal direct sum of irreducible
T -modules. In this paper, each T -module will be considered as a submodule of V ; we
can do this since V is faithful.
An irreducible T -module W is thin if dim(E i∗ W ) ≤ 1 for all i. The endpoint of W is
min{i : E i∗ W 6= 0} (note that this is called the dual endpoint in [4]). There is a unique
irreducible T -module of endpoint 0, called the trivial module; it is thin and has basis
{E i∗ 1 : 0 ≤ i ≤ d}, where 1 is the vector of all 1’s.
Throughout the paper, we will assume X = (X, {Ri }0≤i≤d ) is a P- and Q-polynomial
scheme of diameter d ≥ 3. As usual, we denote the intersection numbers by ai , bi , and
ci , where ai = |0i (x) ∩ 01 (y)|, bi = |0i+1 (x) ∩ 01 (y)|, and ci = |0i−1 (x) ∩ 01 (y)| for
be the eigenvalues
y ∈ 0i (x). We also let k = |01 (x)|, the valencyPof the graph. Let {θi }P
and {θi∗ } the dual eigenvalues of X , so A1 =
θi E i and E 1 = |X1 | θi∗ Ai . It is well
known that there exists a nonzero constant Q such that [1]
(θi+3 − θi+2 ) − ρ(θi+2 − θi+1 ) + (θi+1 − θi ) = 0
∗
∗
∗
∗
∗
− θi+2
) − ρ(θi+2
− θi+1
) + (θi+1
− θi∗ ) = 0
(θi+3
(0 ≤ i ≤ d − 3)
(0 ≤ i ≤ d − 3).
where ρ = q + q −1 .
It is convenient to define θi and θi∗ for all integers i by the above recurrence. Note that
θi (and similarly θi∗ ) are distinct for 0 ≤ i ≤ d, but they may not be so in general.
As in [2], we say that X has classical parameters (d, q, α, β) if X has diameter d and
intersection numbers
µ· ¸ · ¸¶µ
· ¸¶
d
i
i
bi =
−
β −α
(0 ≤ i ≤ d)
1
1
1
· ¸µ
·
¸¶
i
i −1
1+α
(0 ≤ i ≤ d),
ci =
1
1
where [ 1j ] = [ 1j ]q = 1 + q + q 2 + · · · + q j−1 is the usual q-binomial coefficient. In this
case, the eigenvalues and dual eigenvalues of X satisfy
· ¸
i
−i
(0 ≤ i ≤ d)
θi = q bi −
1
· ¸
i 1−i
q
(0 ≤ i ≤ d).
θi∗ = θ0∗ + (θ1∗ − θ0∗ )
1
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Fix x ∈ X , and write T = T (x), E i∗ = E i∗ (x), and Vi∗ = Vi∗ (x). Let
F=
R=
L=
d
X
i=0
d−1
X
i=0
d
X
E i∗ A1 E i∗
(the flat operator)
∗
E i+1
A1 E i∗
(the raise operator)
∗
E i−1
A1 E i∗
(the lower operator).
i=1
Note that F, L, R ∈ T , and A1 = F + L + R. F is a symmetric 0, 1 matrix, and R and
L are 0, 1 matrices such that R t = L.
We will use extensively the following relations on these operators, which were given in
[4] in a slightly different form.
Proposition 2.1 ([4], Lemmas 5.5 and 5.6)
(gi− F L 2 + L F L + gi+ L 2 F − γ L 2 )E i∗
− 2
∗
(gi R F + R F R + gi+ F R 2 − γ R 2 )E i−2
−
+
(ei R L 2 + (ρ + 2)L R L + ei L 2 R + L F 2 − ρ F L F
+ F 2 L − γ (L F + F L) − δL)E i∗
(ei− R 2 L + (ρ + 2)R L R + ei+ L R 2 + F 2 R − ρ F R F
∗
+ R F 2 − γ (F R + R F) − δ R)E i−1
=0
=0
(2 ≤ i ≤ d)
(2 ≤ i ≤ d)
(2.1)
(2.2)
=0
(1 ≤ i ≤ d)
(2.3)
=0
(1 ≤ i ≤ d)
(2.4)
where
∗
∗
∗
θi∗ − θi+1
+ θi+2
− θi+3
∗
∗
θi+1 − θi+2
γ = θi − ρθi+1 + θi+2
ρ = q + q −1 =
δ=
θi2
− ρθi θi+1 +
2
θi+1
− γ (θi + θi+1 )
which are constants independent of i, and
gi− =
∗
∗
θi−2
− θi−3
∗
θi−2
− θi∗
(2 ≤ i ≤ d)
gi+ =
∗
θi∗ − θi+1
∗
∗
θi − θi−2
(2 ≤ i ≤ d)
ei− =
∗
∗
θi−3
− θi−1
∗
θi∗ − θi−1
(1 ≤ i ≤ d)
ei+ =
∗
θi∗ − θi+2
∗
θi∗ − θi−1
(1 ≤ i ≤ d).
(0 ≤ i ≤ d − 3)
(0 ≤ i ≤ d − 2)
(0 ≤ i ≤ d − 1),
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If X has classical parameters (d, q, α, β), the above are given by
ρ = q + q −1
α−β +1
γ = q d−1 α + β − 1 +
q
½·
¸·
¸
µ·
¸ ·
¸¶
1 d +1 d −1 2
d +1
d −1
α −
+
(β − 1)α
δ=
1
1
1
1
q
¾
+ (q + 1)2 q d−1 β + (β − 1)2
−q 2
q +1
−1
gi+ =
q(q + 1)
gi− =
ei− = −q(q + 1)
ei+ =
2.2.
−(q + 1)
.
q2
Irreducible modules of endpoint 1
We now summarize the relevant results of [5] and [7] about irreducible modules of endpoint 1.
Let U1∗ be the subspace of V1∗ which is orthogonal to 1. Note that U1∗ is the subspace of
V1∗ which is orthogonal to the trivial module, and hence E 1∗ T E 1∗ acts on U1∗ .
Theorem 2.2 ([5]) Let W be an irreducible T -module of endpoint 1. Then E 1∗ W is a
one-dimensional subspace of U1∗ . In particular, any nonzero v ∈ E 1∗ W is an eigenvector
of E 1∗ A1 E 1∗ , and W = T v. Conversely, let v ∈ U1∗ be an eigenvector of E 1∗ A1 E 1∗ . Then
T v is an irreducible T -module of endpoint 1.
Theorem 2.3 ([5]) Let v, v 0 ∈ U1∗ be eigenvectors for E 1∗ A1 E 1∗ with corresponding
eigenvalues λ, λ0 . Then T v and T v 0 are isomorphic as T -modules if and only if λ = λ0 .
Let W = T v be an irreducible T -module of endpoint 1, where v is an eigenvector of
E 1∗ A1 E 1∗ acting on U1∗ . Define
vi+ = E i∗ Ai−1 v
vi− = E i∗ Ai+1 v.
+
−
Theorem 2.4 ([5]) The set of vectors {v = v1+ , v2+ , v2− , . . . , vd−1
, vd−1
, vd+ } spans W . It
is a basis for W if W is not thin. In particular, if W is not thin, dim E i∗ W = 2 for
2 ≤ i ≤ d − 1, and dim E i∗ W = 1 for i = 1, d.
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Theorem 2.5 ([7]) Suppose W is a nonthin irreducible T -module of endpoint 1. Then the
following maps are nonsingular.
∗
R| Ei∗ W : E i∗ W → E i+1
W
∗
∗
W
L| Ei∗ W : E i W → E i−1
(2 ≤ i ≤ d − 2)
(3 ≤ i ≤ d − 1)
+
−
The basis {v = v1+ , v2+ , v2− , . . . , vd−1
, vd−1
, vd+ } will be called the Terwilliger basis for W .
The following lemma gives some easy consequences of the definition of these vectors.
+
are all the zero vector.
Note that v0− , v0+ , vd− , and vd+1
Lemma 2.6 ([5]) For 1 ≤ i ≤ d,
+
Rvi+ = ci vi+1
−
Lvi− = bi vi−1
+
∗
E i Ai v = −vi − vi−
−
+ (ai−1 + ci−1 − ci )vi+ − ci vi−
Fvi+ = Rvi−1
−
+
−
+ bi−1 vi−1
+ (ci − ai−1 − ci−1 )vi−1
.
Lvi+ = Fvi−1
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
For classical parameters, the following theorem tells exactly which modules are thin.
Theorem 2.7 ([5]) Suppose X has classical parameters (d, q, α, β). Let v ∈ U1∗ be an
eigenvector for E 1∗ A1 E 1∗ with eigenvalue λ. The irreducible module T v of endpoint 1 is
thin if and only if
½
·
¸
¾
d −1
λ ∈ −1, −q − 1, β − α − 1, αq
−1 .
1
3.
Ladder bases
In this section, we describe a particularly nice sort of basis for a nonthin irreducible module
of endpoint 1, which we will call a ladder basis. We will then give criteria for the existence
of such a basis.
Throughout Sections 3 to 5, we assume that W is a nonthin irreducible T -module of
endpoint 1, and v is a nonzero vector in E 1∗ W . Then v is an eigenvector for E 1∗ A1 E 1∗ , and
we denote the corresponding eigenvalue by λ. By Theorem 2.2, W = T v.
3.1.
Criteria for existence
+
−
A ladder basis for W is an orthogonal basis {w1− , w2+ , w2− , . . . , wd−1
, wd−1
, wd+ } which
satisfies the following.
(i) E i∗ W = span{wi+ , wi− }.
(ii) wi+ , wi− are eigenvectors for F.
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(iii) For 2 ≤ i ≤ d − 1,
+
Rwi+ ∈ span{wi+1
}
−
−
}.
Lwi ∈ span{wi−1
If the eigenvalues of F on E i∗ W are distinct, then the corresponding eigenvectors are
automatically orthogonal. In this case, the following lemma shows that it is enough to
check one of the properties of (iii).
+
,
Lemma 3.1 Suppose i is a fixed integer with 2 ≤ i ≤ d − 2, and wi+ , wi− , wi+1
−
+
−
∗
∗
wi+1 are mutually orthogonal vectors such that Ei W = span{wi , wi } and E i+1 W =
+
−
, wi+1
}.
span{wi+1
−
+
}.
Then Lwi+1 ∈ span{wi− } if and only if Rwi+ ∈ span{wi+1
+
−
−
Proof: Since hwi+ , wi− i = 0 = hwi+1
, wi+1
i, it follows that Lwi+1
∈ span{wi− } if and
−
+
−
+
−
+
only if hLwi+1 , wi i = 0. But hLwi+1 , wi i = hwi+1 , Rwi i, so this occurs if and only if
+
}.
2
Rwi+ ∈ span{wi+1
We can now use this and the relations on the operators F, L, and R to show that we only
need to check a few eigenvectors to see if a ladder basis exists.
Proposition 3.2 The following are equivalent.
(i) W has a ladder basis.
(ii) There exist eigenvectors w2+ and w3+ for F such that w2+ ∈ E 2∗ W, w3+ ∈ E 3∗ W, and
Rw2+ ∈ span{w3+ }.
(iii) There exist eigenvectors w2− and w3− for F such that w2− ∈ E 2∗ W, w3− ∈ E 3∗ W, and
Lw3− ∈ span{w2− }.
Proof:
(ii ⇒ iii): Since F is symmetric, there exist eigenvectors w2− and w3− such that E 2∗ W = span{w2+ ,
w2− }, hw2+ , w2− i = 0, E 3∗ W = span{w3+ , w3− } and hw3+ , w3− i = 0. By Lemma 3.1, Lw3− ∈
span{w2− }.
(iii ⇒ ii): Similar.
(i ⇒ ii): Clear.
+
, 4 ≤ i ≤ d − 1, and let wd+ be any nonzero
(ii ⇒ i): Define wi+ inductively by wi+ = Rwi−1
+
∗
vector of E d W . By Theorem 2.5, wi 6= 0 for i < d.
+
+
+
+
Suppose wi−1
and wi−2
are eigenvectors for F, with eigenvalues λi−1
and λi−2
, respec+
tively. Applying the operator of (2.2) to wi−2 , we find
+
+
gi− λi−2
wi+ + λi−1
wi+ + gi+ Fwi+ − γ wi+ = 0.
Since gi+ is nonzero, Fwi+ ∈ span{wi+ }, and wi+ is an eigenvector for F. Since dim E d∗ W
+
∈ span{wd+ }. So the set {w2+ , w3+ , . . . , wd+ } is
= 1, wd+ is an eigenvector for F and Rwd−1
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half of a ladder basis. Now choose wi− an eigenvector for F so that E i∗ W = span{wi+ , wi− }
and hwi+ , wi− i = 0, and we are finished by Lemma 3.1.
2
3.2.
Eigenvalues of F
+
−
, wd−1
, wd+ } is a ladder basis for W , and that {λi+ } and
Suppose {w1− , w2+ , w2− ,. . . , wd−1
−
{λi } are the corresponding eigenvalues, so Fwi+ = λi+ wi+ , and Fwi− = λi− wi− . Note that
λ−
1 = λ. For convenience, we will define
+ +
γ − λ+
2 − g3 λ3
,
g3−
λ+
1 =
where γ , g3+ , and g3− are as in Proposition 2.1.
Theorem 3.3 The eigenvalues λi+ and λi− (3 ≤ i ≤ d − 1) of F are given in terms of λ+
1,
−
−
λ+
2 , λ1 , and λ2 as follows:
−(γ (θi∗ − θ2∗ )(θi∗ − θ1∗ ) − λ2 (θ3∗ − θ2∗ )(θi∗ − θ1∗ ) + λ1 (θ1∗ − θ0∗ )(θi∗ − θ2∗ ))
∗
∗
(θi+1
− θi∗ )(θi∗ − θi−1
)
(3.1)
λi =
where either
(+)
λi = λi+
(2 ≤ i ≤ d − 1),
(−)
λi = λi−
(2 ≤ i ≤ d − 1).
or
If gd+ 6= 0, then this formula applies for λ+
d also.
Proof of Theorem 3.3: Note that in any case we can find λ+
d using tr(A1 |W ) =
Pd−1
P −
λi = θ1 +2 i=2 θi + θd .
P
λi+ +
2
Proof: The equation
gi− λi−2 + λi−1 + gi+ λi − γ = 0
(3 ≤ i ≤ d)
(3.2)
+
; for i = 3, it holds
holds in either case. To see this for (+), i ≥ 4, apply (2.2) to wi−2
+
−
+
by definition of λ1 . For (−), apply (2.1) to wi . If gi 6= 0 (which holds at least for
2 ≤ i ≤ d − 1), this determines λi recursively.
It is easy to check that (3.1) is a solution to this equation. However, we will give below
a method for solving the recursion.
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Substitute the values for gi− and gi+ into (3.2); this results in the equation
∗
∗
∗
∗
∗
∗
((θi+1
− θi∗ )λi − (θi−1
− θi−2
)λi−1 ) − ((θi∗ − θi−1
)λi−1 − (θi−2
− θi−3
)λi−2 )
∗
)γ = 0.
+ (θi∗ − θi−2
(3.3)
Summing (3.3) for i = 3 to j,
(θ ∗j+1 − θ ∗j )λ j − (θ ∗j−1 − θ ∗j−2 )λ j−1 − (θ3∗ − θ2∗ )λ2 + (θ1∗ − θ0∗ )λ1
+ γ (θ ∗j + θ ∗j−1 − θ2∗ − θ1∗ ) = 0,
(3.4)
which holds for 2 ≤ j ≤ d.
Define µ j by
λj =
µj
(θ ∗j+1 − θ ∗j )(θ ∗j − θ ∗j−1 )
and let
σ = −(θ3∗ − θ2∗ )λ2 + (θ1∗ − θ0∗ )λ1 − γ (θ2∗ + θ1∗ ),
so (3.4) becomes
¡
¢
∗
∗
µ j − µ j−1 + γ θ ∗j 2 − θ ∗2
j−1 + σ (θ j − θ j−1 ) = 0.
(3.5)
Summing (3.5) from 2 to i results in
¡
¢
µi − µ1 + γ θi∗ 2 − θ1∗ 2 + σ (θi∗ − θ1∗ ) = 0,
from which the theorem follows.
4.
2
The Terwilliger basis
+
−
, vd−1
, vd+ } of W is defined by
Recall that the Terwilliger basis {v = v1+ , v2+ , v2− , . . . , vd−1
+
−
−
∗
∗
vi = E i Ai−1 v, vi = E i Ai+1 v. Occasionally it is useful to let v1 = E 1∗ A2 v and vd− = 0.
It follows from (2.7) that v1− = (−λ − 1)v.
In this section, we find explicitly the action of A1 on vi+ and vi− , and show that an
association scheme with classical parameters must have a ladder basis.
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4.1.
HOBART AND ITO
Notation
We will use the following notation for entries of A1 |W .
+
−
Rvi− = ri+ vi+1
+ ri− vi+1
−
Rvd−1
Lvi+
Lv2+
Fvi+
Fvi−
Fvd+
=
=
=
=
=
=
+
rd−1
vd+
+
−
li+ vi−1
+ li− vi−1
l2+ v1+
f i++ vi+ + f i−+ vi−
f i+− vi+ + f i−− vi−
f d++ vd+
(2 ≤ i ≤ d − 2)
(3 ≤ i ≤ d)
(2 ≤ i ≤ d − 1)
(2 ≤ i ≤ d − 1)
Note that Fv1+ = λv1+ , and from (2.6), Lv2− = b2 v1− = (−λ − 1)b2 v1+ . We know Rvi+ and
Lvi− from Lemma 2.6.
Thus, with respect to the Terwilliger basis, A1 |W has matrix

λ
1


0

















Lemma 4.1
l2+
f 2++
f 2−+
c2
0
−(λ + 1)b2
f 2+−
f 2−−
r2+
r2−

l3+
l3−
f 3++
f 3−+
0
b3
f 3+−
f 3−−
l4+
l4−
0
b4
..
..
..
.
.
.
..
.
++
f d−1
−+
f d−1
cd−1
+−
f d−1
−−
f d−1
+
rd−1
ld+
ld−
f d++










.










For 2 ≤ i ≤ d − 1,
+
f i+− = li+1
− bi
−−
−
f i = li+1 + ai + ci − ci+1
+
+ ai−1 + ci−1 − ci
f i++ = ri−1
−+
−
− ci , where r1− = 0.
f i = ri−1
Proof: This follows directly from (2.8) and (2.9).
(4.1)
(4.2)
(4.3)
(4.4)
2
The remainder of this section will be restricted to the case of classical parameters. The
methods here clearly give formulae in the general P- and Q-polynomial case for the entries
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of A1 |W , however they are quite complicated. We expect to consider the general case using
a different approach in a subsequent paper.
4.2.
Showing a ladder basis exists
Theorem 4.2 Suppose X has classical parameters. Then fi+− = 0, and hence vi− is an
eigenvector for F, 2 ≤ i ≤ d − 1.
Proof: We first show that f 2+− and f 3+− are equal to 0.
Most of the calculations use Eqs. (2.1) to (2.4) on the operators F, L, and R.
From (2.9) with i = 2 and the fact that v1− = −(λ + 1)v1+ , we find
l2+ = (λ + 1)(a1 − c2 − λ + 1) + b1 .
(4.5)
Since X is P-polynomial, we can write
A2 =
¢
1 ¡ 2
A 1 − k I − a1 A 1
c2
and this together with (2.7) results in
f 2++ = a1 − c2 − λ
f 2−+ = −c2 .
(4.6)
(4.7)
Now apply (2.4) to v1+ , and consider coefficients of v2+ . This results in the equation
(ρ + 2)l2+ + e2+ c2l3+ + f 2++ + f 2−+ f 2+− − ρλ f 2++ + λ2 − γ ( f 2++ + λ) − δ = 0
2
which we may consider as being linear in the unknowns f 2+− and l3+ ; everything else
is known as a function of q, d, α, and β. Solving simultaneously with the equation
f 2+− − l3+ + b2 = 0 from (4.1), we find
f 2+− = 0
l3+ = b2 .
(4.8)
(4.9)
Similarly, apply (2.4) to v1+ and consider the coefficients of v2− to get a linear equation
in l3− and f 2−−
e2+ c2l3− + f 2++ f 2−+ + f 2−+ f 2−− − ρλ f 2−+ − γ f 2−+ = 0.
Solving simultaneously with f 2−− − l3− − a2 − c2 + c3 = 0 from (4.2), we find
f 2−− = q(λ − a1 ) + a2
l3− = q(λ − a1 ) + c3 − c2 .
(4.10)
(4.11)
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Apply (2.2) to v1+ , and consider the coefficient of v3+ resulting in the equation
g3− c2 λ + f 2++ c2 + f 2−+r2+ + g3+ c2 f 3++ − γ c2 = 0
which may be solved simultaneously with
f 3++ − r2+ + c3 − a2 − c2 = 0,
and the coefficient of v3− resulting in the equation
f 2−+r2− + g3+ c2 f 3−+ = 0
which may be solved simultaneously with
f 3−+ − r2− + c3 = 0.
The results are
k − b3
− c3
q2 + q + 1
= −(λ + 1)q + a2 + c2 − c3
k − b3
+
r2 = q(a1 − λ) − k + b2 + 2
= −(λ + 1)q
q +q +1
−q 2 − q
c3
f 3−+ = 2
q +q +1
c3
r2− = 2
.
q +q +1
f 3++ = q(a1 − λ) +
(4.12)
(4.13)
(4.14)
(4.15)
Finally, apply (2.4) to v2+ and consider the coefficient of v3+ to obtain
¡
¢
2
0 = e3−l2+ c2 + (ρ + 2) l3+ c22 + l3− c2r2+ + e3+ c2 c3l4+ + c2 f 3++ + c2 f 3−+ f 3+−
¡
¢
2
− ρ( f 2++ f 3++ c2 + f 3++ f 2−+r2+ + f 3+− f 2−+r2− ) + c2 f 2++ + f 2−+ f 2+−
+ r2+ f 2−+ ( f 2++ + f 2−− ) − γ (c2 f 3++ + c2 f 2++ + f 2−+r2+ ) − δc2 .
Solving this simultaneously with the equation
f 3+− − l4+ + b3 = 0,
we find
f 3+− = 0
l4+ = b3 .
Now we can show f i+− = 0 for all i.
(4.16)
(4.17)
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+
Apply (2.1) to vi− , 4 ≤ i ≤ d − 1, and consider the coefficients of vi−2
. The resulting
equation is
+−
+− +
+
+ bi f i−1
li−1 + gi+ f i+−li+li−1
=0
gi− bi bi−1 f i−2
(4 ≤ i ≤ d − 1).
+−
+ bi−1 . Now inductively, starting with f 2+− = f 3+− = 0
Note also that by (4.1), li+ = f i−1
+
+
+
= bi ).
2
and hence l3 = b2 , l4 = b3 , we can show that f i+− = 0 (and hence li+1
Corollary 4.3 If X has classical parameters, then W has a ladder basis.
Proof: By Theorem 4.2, v2− and v3− are eigenvectors with Lv3− ∈ span{v2− }. Now by
Proposition 3.2, W has a ladder basis.
2
5.
The entries of the matrix
In this section, we give our computational results about the entries of the matrix for A1 |W
with respect to both the Terwilliger and ladder bases.
5.1.
The matrix with respect to the Terwilliger basis
Theorem 5.1 Suppose X has classical parameters. Then the entries of A1 |W with respect
to the Terwilliger basis, using the notation of Section 4.1, are given by
l2+ = (λ + 1)(a1 − c2 − λ + 1) + b1
£ i−2 ¤
f i++ = q i−2 (a1 − λ) + £ 1i ¤ (k − bi ) − ci
(5.1)
(2 ≤ i ≤ d)
(5.2)
f i−− = q i−1 (λ − a1 ) + ai
(2 ≤ i ≤ d − 1)
(5.3)
f i+−
(2 ≤ i ≤ d − 1)
(5.4)
(2 ≤ i ≤ d − 1)
(5.5)
(3 ≤ i ≤ d)
(5.6)
(3 ≤ i ≤ d)
(5.7)
1
=0
à £ i−2 ¤
f i−+ =
!
£ 1i ¤ − 1 ci
1
li+ = bi−1
li− = q i−2 (λ − a1 ) + ci − ci−1
£ i−1 ¤
1
ri+ = q i−1 (a1 − λ) − k + bi + £ i+1
¤ (k − bi+1 ) (2 ≤ i ≤ d − 1)
ri−
£ i−1 ¤
1
= £ i+1
¤ ci+1
(5.8)
1
(2 ≤ i ≤ d − 2).
(5.9)
1
Proof: We will make extensive use of the intermediate calculations in the proof of
Theorem 4.2.
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The formula (5.1) is just (4.5).
To find (5.2) and (5.3), we apply Theorem 3.3; we can do this since W has a ladder basis,
−
and E i∗ W has eigenvalues f i++ and f i−− . Here, λ−
1 = λ, and λ2 = q(λ − a1 ) + a2 , hence
f i−− = λi−
=−
−
=−
(θ3∗ − θ2∗ )(θi∗ − θ1∗ )
(θi∗ − θ2∗ )(θi∗ − θ1∗ )γ
+
(q(λ − a1 ) + a2 )
∗
∗
∗
∗
(θi+1
− θi∗ )(θi∗ − θi−1
) (θi+1
− θi∗ )(θi∗ − θi−1
)
(θ1∗ − θ0∗ )(θi∗ − θ2∗ )
λ
∗
∗
(θi+1
− θi∗ )(θi∗ − θi−1
)
q i (q i − q)
q i (q i − q 2 )
(q i − q)(q i − q 2 )
γ
+
)
+
a
)
−
(q(λ
−
a
λ
1
2
q 2 (q − 1)2
q 3 (q − 1)
q 2 (q − 1)
and using the values of γ , a1 , and a2 for classical parameters, we can check that this equals
(5.3). Using (4.2), we also get (5.7).
f i++ = λi+ can be calculated similarly. From (4.6), f 2++ = λ+
2 = a1 − c2 − λ and
+
+
+ +
−
from (4.12), f 3++ = λ+
3 = −(λ + 1)q + a2 + c2 − c3 , so λ1 = (γ − λ2 − g3 λ3 )/g3 =
−(λ + 1 + q)/q. Now (5.2) follows from Theorem 3.3.
Using (4.3), we also get (5.8) for i ≤ d − 2; the case i = d − 1 can be checked directly
using (2.8).
+
, 3 ≤ i ≤ d − 2,
To show (5.9), we use another recursive equation. Apply (2.2) to vi−1
−
and consider the coefficients of vi+1 . The resulting equation is
−
−+ − −
+
−+
f i−1
ri−1ri + ci−1 f i−+ri− + gi+1
ci ci−1 f i+1
= 0.
gi+1
−
+
By (4.4), f i−+ = ri−1
− ci for i ≥ 2. Hence substituting this and the values of gi+1
and
−
−
gi+1 for classical parameters and solving for ri , we have for 3 ≤ i ≤ d − 1
ri− =
−
q 3 (ri−2
−
−
ci−1 )ri−1
ci−1 ci ci+1
.
−
+ ci ci−1 − q(q + 1)(ri−1
− ci )ci−1
It is easily checked that (5.9) satisfies this equation.
(5.10)
2
Corollary 5.2 Suppose X has classical parameters. For 2 ≤ i ≤ d − 1,
i. f i−+ 6= 0
ii. f i++ − f i−− 6= 0.
Proof: Since ci 6= 0, it follows from (5.5) that f i−+ 6= 0. F is symmetric, and hence F| Ei∗ W
is diagonalizable, and has eigenvalues f i++ and f i−− by Theorem 4.2. If f i++ = f i−− , this
2
implies f i−+ = 0, a contradiction.
We can also write all entries in terms of the d, q, α, β, and λ.
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Corollary 5.3 If X has classical parameters (d, q, α, β), then
½µ· ¸
¶
¾
· ¸
d
d
− q − 2 α + β − q − 2 λ − (q + 1)α +
β −q −1
1
1
µ ·
¸ ·
¸¶ ·
¸
·
¸
i −1
i −1
i −1
i d −i
= q
−
α+
(β − 1) − q i−2 (λ + q + 1)
1
1
1
1
µ ·
¸ ·
¸¶ ·
¸
·
¸
i +1
i −1
i −1
i d −i
= q
−
α+
(β − 1) + q i−1 λ
1
1
1
1
l2+ = −λ2 +
f i++
f i−−
f i+− = 0
µ ·
¸
¶
i −1
f i−+ = −q i−2 (q + 1) α
+1
1
·
¸µ
·
¸¶
d −i +1
i −1
β −α
for i ≥ 3
li+ = q i−1
1
1
½ µ·
¸
·
¸¶
¾
i −1
d −i
− qi
α−β +λ+q +1
li− = q i−2
1
1
ri+ = −q i−1 (λ + 1)
·
¸µ
· ¸ ¶
i −1
i
ri− =
1+
α .
1
1
5.2.
The matrix with respect to the ladder basis
We will assume that X has classical parameters (d, q, α, β). By Corollary 5.2, f i++ 6= f i−− ,
and the eigenvectors of F|W are orthogonal. Let
wi− = vi−
µ
¶
(1 ≤ i ≤ d − 1)
f i++ − f i−− +
vi + vi−
(2 ≤ i ≤ d − 1)
−+
f
i
1
Ã
£ d−1 ¤ !
cd λ + 1 − q 1 α +
+
vd .
wd = £ d ¤
£
¤
1 + d−1
α
1
1
ci
wi+ = £ i ¤
Note that
£ ¤
α
λ + 1 − q i−1
f i++ − f i−−
,
=
£ i−1 ¤1
−+
fi
1+ 1 α
so (since vd− = 0), wd+ follows the same formula as the other vectors wi+ . By Theorem 2.7,
wd+ 6= 0.
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It is easily checked that wi+ and wi− are eigenvectors for F, and that
−
Lwi− = bi wi−1
.
+
−
By Lemma 3.1, Rwi+ ∈ span{wi+1
}. The coefficient of vi+1
in Rwi+ is
£ i−1 ¤
ci+1 ci
ri− ci
£ i ¤ = £1i+1 ¤£ i ¤ ,
1
1
hence
Rwi+
1
£ i−1 ¤
+
= £ 1i ¤ ci wi+1
.
1
{w−1 ,
Therefore
w+2 , w−2 , . . . , w+d−1 , w−d−1 , w+d } is a ladder basis for X .
For this basis, we use the notation
+
−
+ σi− wi−1
Lwi+ = σi+ wi−1
Lw2+
Rwi−
−
Rwd−1
=
=
=
σ2− w1−
+
−
τi+ wi+1
+ τi− wi+1
+
τd−1
wd+ .
(3 ≤ i ≤ d)
(1 ≤ i ≤ d − 2)
So A1 |W has matrix with respect to this basis given by

λ−
1
τ+
 1
 −
 τ1
















σ2−
λ+
2
0
c2
q+1
0
b2
0
λ−
2
τ2+
τ2−

σ3+
σ3−
λ+
3
0
0
b3
0
λ−
3
σ4+
σ4−
0
b4
..
.
..
..
.
0
£ d−2 ¤
1
£ d−1
¤ cd−1
1
Now, using Theorem 5.1, and the equations
µ£ i ¤
¶
f i−+
+
+
−
1
w − wi
vi = ++
f i − f i−− ci i
vi− = wi− ,
we can easily calculate the entries of this matrix.
.
λ+
d−1
0
λ−
d−1
+
τd−1










.


+
σd 

σd− 



λ+
d
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Theorem 5.4 With notation as above,
µ ·
¸ ·
¸¶ ·
¸
·
¸
i −1
i −1
i −1
+
i d −i
−
α+
(β − 1) − q i−2 (λ + q + 1)
λi = q
1
1
1
1
λi−
σi+
σi−
6.
(2 ≤ i ≤ d)
µ ·
¸ ·
¸¶ ·
¸
·
¸
d −i
i +1
i −1
i −1
= qi
−
α+
(β − 1) + q i−1 λ
1
1
1
1
(1 ≤ i ≤ d − 1)
£ i−1 ¤ ¢¡
£ i−1 ¤ ¢
β − 1 α λ+1−q 1 α
q
1
(3 ≤ i ≤ d)
=
¡
£ ¤ ¢
λ + 1 − q i−2
α
1
¡
£
¤ ¢
α
q i−2 (q + λ + 1)(λ + 1 + α − β) λ + 1 − q d−1
1
(2 ≤ i ≤ d)
=
¡
£ i−2 ¤ ¢
λ+1−q 1 α
£
¤¡
i−1 d+1−i
q i−1 (λ + 1)
£i ¤
q 1 α − (λ + 1)
£ i ¤¡£ i ¤
¢¡
£ ¤ ¢
α + 1 λ + 1 − q i−1
α
1
1
1
=
£i ¤
λ+1−q 1 α
τi+ =
(1 ≤ i ≤ d − 1)
τi−
(1 ≤ i ≤ d − 2).
Examples
The irreducible modules of endpoint 1 are determined up to isomorphism by the eigenvalues
of the graph on the first subconstituent 01 (x), since E 1∗ A1 E 1∗ is the adjacency matrix of this
graph. In this section, we consider three families of P- and Q-polynomial schemes which
are not thin: the bilinear forms schemes, the Hermitean forms schemes, and the alternating
forms schemes. For each, we find the eigenvalues of 01 (x) and then use the results of
Section 5 to find the entries of A1 |W for each nonthin irreducible module W of endpoint 1.
We use [2] as a reference for these schemes.
The only other known P- and Q-polynomial schemes of diameter ≥6 which are not thin
are the Doob schemes and the quadratic forms schemes [4]. All irreducible modules for the
Doob schemes were determined by Tanabe [3]. The case of the quadratic forms schemes
appears to be much more difficult.
6.1.
The bilinear forms schemes
Let X be the set of d × n matrices over G F(q), d ≤ n. Define matrices M and N to have
relation i if rk(M − N ) = i. This gives a P- and Q-polynomial scheme, called the bilinear
forms scheme, which has classical parameter (d, q, α, β) with α = q − 1 and β = q n − 1.
The automorphism group acts transitively on this scheme, therefore the structure of T (x)
is independent of x.
Assume d ≥ 3.
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Theorem 6.1 The induced subgraph on 01 (x) for the bilinear forms scheme has eigenvalues and multiplicities as shown below. We also show whether the corresponding module
is the trivial module (of endpoint 0), or a thin or nonthin irreducible module of endpoint 1.
Note that two of the eigenvalues coincide if d = n, and that −1 does not occur as an
eigenvalue if q = 2.
eigenvalue
multiplicity
corresponding module
1
trivial
qd + qn − q − 2
qn − q − 1
q d −q
q−1
thin
q −q −1
q n −q
q−1
thin
−1
(q d −1)(q n −1)(q−2)
(q−1)2
thin
−q
(q d −q)(q n −q)
(q−1)2
nonthin
d
Proof: We may assume x = 0, the d × n zero matrix, so 01 (x) has as vertices the set of
all d × n matrices of rank 1. Any rank 1 matrix M can be written uniquely as M = µuv t ,
where µ ∈ G F(q)∗ , and u and v are d and n-tuples respectively with first nonzero entry
equal to 1.
Suppose M = µuv t . Clearly M is adjacent to µ1 u 1 v1t if u = u 1 or v = v1 . The number
of such matrices is equal to a1 , hence this characterizes the matrices adjacent to M. (This
could also be easily checked using the distance-transitivity of the automorphism group and
suitable u and v.)
Given any such u and v, {µuv t | µ ∈ G F(q)∗ } forms a clique in 01 (x), and the vertices
of two such cliques are either all adjacent or all nonadjacent. We can therefore form a
d
n
−1)
quotient graph on the (q −1)(q
cliques of this form. The quotient graph has adjacency
(q−1)2
matrix
B = (J − I ) ⊗ I + I ⊗ (J − I ).
This matrix has eigenvalues
q n −q
,
q−1
q n +q d −2q q n −1
, q−1
q−1
− 2,
q d −1
q−1
− 2, and −2 of multiplicity 1,
q d −q
,
q−1
−q)
and (q −q)(q
, respectively.
(q−1)2
Now 01 (x) has adjacency matrix B ⊗ Jq−1 + I ⊗ (J − I ), and the result follows.
d
n
2
Suppose W is the unique nonthin irreducible module. Using Theorems 5.1 and 5.4, we
can find the matrix of A1 |W with respect to either the Terwilliger basis or the ladder basis
of Section 5.2.
The eigenvalues of F on E i∗ W are given by
·
¸
· ¸
i −1
i i−2
(q d + q n − q i − 1) −
q
λi+ = f i++ =
1
1
·
¸
· ¸
i −1
i i
−
−−
d
n
i
λi = f i =
(q + q − q − 1) −
q
1
1
where λ = λ−
1.
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71
For the Terwilliger basis, the other entries of the matrix of A1 |W are
(q d − q 2 + q − 1)(q n − q 2 + q − 1)
+ q2 − q
q −1
=0
= −(q + 1)q 2i−3
l2+ =
f i+−
f i−+
(q n − q i−1 )(q d − q i−1 )
q −1
−
i−2 n
li = −q (q + q d − q i − q i−1 − 1)
li+ =
ri+ = q i−1 (q − 1)
·
¸
−
i i −1
ri = q
.
1
For the ladder basis given in Section 5.2, the other entries of A1 |W are
6.2.
σi+ =
(q d − q i−1 )(q n − q i−1 )(q i − 1)
(q − 1)(q i−1 − 1)
σi− =
−q i−1 (q d − 1)(q n − 1)
(q i − q)
τi+ =
−q i−1 (q − 1)
q i+1 − 1
τi− =
q i (q i − 1)2
.
(q − 1)(q i+1 − 1)
The Hermitean forms schemes
Let X be the set of d ×d Hermitean matrices over G F(r 2 ), where r is a prime power. Define
matrices M and N to have relation i if rk(M − N ) = i. This gives a P- and Q-polynomial
scheme, called the Hermitean forms scheme, which has classical paramters (d, q, α, β),
with q = −r , α = −r − 1, and β = −(−r )d − 1.
The automorphism group of this scheme acts transitively, therefore the structure of T (x)
is independent of the choice of x.
Assume d ≥ 3.
Theorem 6.2 The induced subgraph on 01 (x) for the Hermitean forms scheme has eigenvalues and multiplicities as shown below. We also show whether the corresponding module
is the trivial module (of endpoint 0), or a thin or nonthin irreducible module of endpoint 1.
Note that if r = 2, −1 does not occur as an eigenvalue.
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eigenvalue
r −2
multiplicity
1
corresponding module
trivial
r −2
r 2d −r 2
r 2 −1
nonthin
−1
(r −1)(r −2)
r 2 −1
thin
2d
Proof: It follows immediately from [6], Theorem 2.12, that 01 (x) is a disjoint union of
r 2d −1
cliques of size r − 1.
2
r 2 −1
As in Section 6.1, we now give the entries of the matrix A1 |W for the Terwilliger basis
and ladder basis, in terms of q and d. Recall that q = −r .
The eigenvalues of F on E i∗ W are
·
¸
·
¸
i − 1 i−1
i −2
+
++
q (q + 1) −
λi = f i = −
1
1
· ¸
i
(q i + q i−1 + 1)
λi− = f i−− = −
1
where λ = λ−
1.
The other entries for the Terwilliger basis are
−(q 2d − q 4 )
+ q2 − 1
q −1
=0
= −(q + 1)q 2i−3
l2+ =
f i+−
f i−+
li+ =
−(q d − q i−1 )(q d + q i−1 )
q −1
li− = q i−2 (q i + q i−1 − 1)
ri+ = q i−1 (q + 1)
·
¸
i −1
ri− = q i
.
1
The other entries for the ladder basis given in Section 5.2 are
σi+ =
−(q i + 1)(q 2d − q 2i−2 )
(q − 1)(q i−1 + 1)
σi− =
−q i−2 (q 2d − 1)
q i−1 + 1
τi+ =
−q i−1 (q + 1)
q i+1 + 1
τi− =
q i (q i − 1)(q i + 1)
.
(q − 1)(q i+1 + 1)
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6.3.
The alternating forms schemes
Let X be the set of n × n skew-symmetric matrices with 0 diagonal over G F(r ); note that
such matrices have even rank. Two matrices M and N have relation i if rk(M − N ) = 2i.
This defines the alternating forms scheme, which is a P- and Q-polynomial scheme with
classical parameters d = bn/2c, q = r 2 , α = r 2 − 1, and β = r m − 1, where m = 2d − 1
if n = 2d, and m = 2d + 1 if n = 2d + 1.
Since the automorphism group is transitive, the structure of the Terwilliger algebra T (x)
for the scheme is independent of the choice of x.
Assume d ≥ 3.
The following lemma about ranks follows from some easy matrix theory; or see [2],
Lemmas 9.5.4 and 9.5.5.
Lemma 6.3 Let M1 and M2 be skew symmetric matrices of rank m 1 and m 2 , respectively.
(i) If ker(M1 ) + ker(M2 ) = G F(r )n , then ker(M1 − M2 ) = ker(M1 ) ∩ ker(M2 ).
(ii) rk(M1 − M2 ) = m 1 + m 2 if and only if G F(r )n = ker(M1 ) + ker(M2 ).
Theorem 6.4 The induced subgraph on 01 (x) for the alternating forms scheme has eigenvalues and multiplicities as shown below. We also show whether the corresponding module
is the trivial module (of endpoint 0), or a thin or nonthin irreducible module of endpoint 1.
Note that if r = 2, then −1 does not occur as an eigenvalue.
eigenvalue
multiplicity
corresponding module
n
n−1
2
−r −2
1
trivial
r +r
r n−1 − r 2 − 1
r n −r
r −1
−1
(r −2)(r −1)(r −1)
(r 2 −1)(r −1)
thin
−r 2 + r − 1
(r −1)(r −r )
(r 2 −1)(r −1)
nonthin
n
n
thin
n−1
n−1
2
Proof: Since the automorphism group is transitive, we can assume x = 0, the zero matrix.
Thus the vertices of 01 (x) are the matrices of X of rank 2.
If M and N are two distinct such matrices, rk(M − N ) = 2 or 4. By Lemma 6.3,
rk(M − N ) = 4, and M and N are not adjacent, if and only if G F(r )n = ker(M) + ker(N ).
Thus M and N are adjacent if and only if dim(ker(M) + ker(N )) < n. Given an n − 2
dimensional subspace S, the set of matrices in X with kernel S forms a clique of size r − 1
in 01 (x) and the vertices of two such cliques are either all adjacent or all nonadjacent.
As in the proof of Theorem 6.1, we now form the quotient graph on the [ n2 ]r cliques,
identifying each with the subspace which is the common kernel. This graph has as vertices
the subspace of G F(r )n of dimension n −2, and two subspaces are adjacent if they intersect
in a subspace of dimension n − 3. But this is isomorphic to the r -Johnson graph on the
n−2
−1) r 2 (r n−3 −1)
two-dimensional subspaces of G F(r )n , and has as eigenvalues r (r +1)(r
, r −1 − 1,
r
−1
n
n−1
2
n
−r )
−r
and −r − 1, with multiplicities 1, rr −1
, and (r (r−1)(r
,
respectively.
2 −1)(r −1)
If the quotient graph has adjacency matrix B, then 01 (x) has adjacency matrix B ⊗ Jr −1 +
I ⊗ (J − I ), and the result follows.
2
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Below are the entries of A1 |W with respect to both bases. Recall that [ n2 ]r 2 = 1 + r 2 + · · ·
+r 2(i−1) and q = r 2 ; we use this notation here to show the similarities with the previous
two examples. Also, note that {2d, m} = {n, n − 1} for both even and odd n, and this allows
us to treat both the even and odd cases together.
The eigenvalues of F on E i∗ W are
·
¸
·
¸
i −1
2i − 2
+
++
n
n−1
2i
(r + r
−r )−
− r 2i−3
λi = f i =
1 r2
1
r2
·
¸
· ¸
i −1
2i
λi− = f i−− =
(r n + r n−1 − r 2i ) −
+ r 2i−1
1 r2
1 r2
where λ = λ−
1.
The other entries for the Terwilliger basis are
l2+ =
(r n − r 4 )(r n−1 − r 4 )
+ r n+1 + r n − r 5 − r 2
r2 − 1
f i+− = 0
f i−+ = −r 4i−6 (r 2 + 1)
li+ =
(r n − r 2i−2 )(r n−1 − r 2i−2 )
r2 − 1
li− = −r 2i−4 (r n + r n−1 − r 2i − r 2i−2 − r )
ri+ = r 2i−1 (r − 1)
·
¸
−
2i i − 1
.
ri = r
1 r2
The other entries for the ladder basis given in Section 5.2 are
σi+ =
(r 2i−1 − 1)(r n − r 2i−2 )(r n−1 − r 2i−2 )
(r 2 − 1)(r 2i−3 − 1)
σi− =
−r 2i−2 (r n−1 − 1)(r n−2 − 1)
r 2i−3 − 1
τi+ =
−r 2i−2 (r − 1)
r 2i+1 − 1
τi− =
r 2i (r 2i − 1)(r 2i−1 − 1)
.
(r 2 − 1)(r 2i+1 − 1)
Acknowledgments
Both authors were visitors at Kyushu University, Fukuoka, Japan during the time of this
research, and they would like to thank the Department of Mathematics of that university
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for their hospitality. They would also like to thank Paul Terwilliger for making his lecture
notes available, and for many helpful discussions.
References
1. E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, The Benjamin/Cummings Publishing
Co., 1984.
2. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, 1989.
3. K. Tanabe, “The irreducible modules of the Terwilliger algebras of Doob schemes,” J. Alg. Combin. 6 (1997),
173–195.
4. P. Terwilliger, “The subconstituent algebra of an association scheme,” I J. Alg. Combin. 1 (1992), 363–388; II
J. Alg. Combin. 2 (1993), 73–103; III J. Alg. Combin. 2 (1993), 177–210.
5. P. Terwilliger, The Subconstituent Algebra of a Graph, the Thin Condition, and the Q-Polynomial Property.
Unpublished lecture notes, 1992.
6. P. Terwilliger, “Kite-free distance regular graphs,” Europ. J. Combin. 16 (1995), 405–414.
7. P. Terwilliger, Personal communication.