ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 17 (2019) 185-202
https://doi.org/10.26493/1855-3974.1559.390
(Also available at http://amc-journal.eu)
A diagram associated with the subconstituent algebra of a distance-regular graph
Supalak Sumalroj *
Department of Mathematics, Naresuan University, Phitsanulok, Thailand
Received 20 December 2017, accepted 6 June 2019, published online 18 September 2019
In this paper we consider a distance-regular graph r. Fix a vertex x of r and consider the corresponding subconstituent algebra T = T(x). The algebra T is the C-algebra generated by the Bose-Mesner algebra M of r and the dual Bose-Mesner algebra M* of r with respect to x. We consider the subspaces M, M *, MM *,M *M, MM * M, M *MM *,... along with their intersections and sums. In our notation, MM * means Span{RS | R e M,S e M*}, and so on. We introduce a diagram that describes how these subspaces are related. We describe in detail that part of the diagram up to MM * + M *M. For each subspace U shown in this part of the diagram, we display an orthogonal basis for U along with the dimension of U. For an edge U C W from this part of the diagram, we display an orthogonal basis for the orthogonal complement of U in W along with the dimension of this orthogonal complement.
Keywords: Subconstituent algebra, Terwilliger algebra, distance-regular graph. Math. Subj. Class.: 05E30
1 Introduction
In this paper we consider a distance-regular graph r. Fix a vertex x of r and consider the corresponding subconstituent algebra (or Terwilliger algebra) T = T(x) [32]. The algebra T is the C-algebra generated by the Bose-Mesner algebra M of r and the dual Bose-Mesner algebra M* of r with respect to x. The algebra T is finite-dimensional and semisimple [32]. So it is natural to study the irreducible T-modules. These modules are used in the study of hypercubes [14, 26], dual polar graphs [20, 38], spin models [6, 10],
*The author would like to thank Professor Paul Terwilliger for many valuable ideas and insightful suggestions on my work. This paper was written while the author was an Honorary Fellow at the University of Wisconsin-Madison supported by the Development and Promotion of Science and Technology Talents (DPST) Project, Thailand.
E-mail address: supalaks@nu.ac.th (Supalak Sumalroj)
Abstract
©® This work is licensed under https://creativecommons.org/licenses/by/4.0/
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codes [13, 28], the bipartite property [4, 5, 9, 16, 21, 22, 23, 25, 27], the almost-bipartite property [3, 8, 17], the Q-polynomial property [5, 7, 11, 12, 18, 19, 27, 35], and the thin property [15, 24, 30, 31, 33, 34, 36, 37].
In this paper we discuss the algebra T using a different approach. We consider the subspaces M, M*, MM*, M*M, MM*M, M*MM*,... along with their intersections and sums; see Figure 1. We describe the diagram of Figure 1 up to MM *+M *M. For each subspace U shown in this part of the diagram, we display an orthogonal basis for U along with the dimension of U. For an edge U C W from this part of the diagram, we display an orthogonal basis for the orthogonal complement of U in W along with the dimension of this orthogonal complement. Our main results are summarized in Theorems 6.1 and 6.2. In the last part of the paper we summarize what is known about the part of diagram above MM * + M *M, and we give some open problems.
2 Preliminaries
In this section we recall some facts about distance-regular graphs. We will use the following notation. Let X denote a nonempty finite set. Let MatX (C) denote the C-algebra consisting of the matrices whose rows and columns are indexed by X and whose entries are in C. For B e MatX(C) let B, Bf', and tr(B) denote the complex conjugate, the transpose, and the trace of B, respectively. We endow MatX (C) with the Hermitean inner product ( , ) such that (R, S) = tr(R4S) for all R, S e Matx (C). The inner product ( , ) is positive definite. Let U, V denote subspaces of MatX (C) such that U C V. The orthogonal complement of U in V is defined by UL = {v e V | (v, u) =0 for all u e U}.
Let r = (X, E) denote a finite, undirected, connected graph, without loops or multiple edges, with vertex set X and edge set E. Let d denote the shortest path-length distance function for r. Define the diameter D := max{d(x, y) | x, y e X}. For a vertex x e X and an integer i > 0 define Tj(x) = {y e X | d(x, y) = i}. For notational convenience abbreviate r(x) = ri(x). For an integer k > 0, we say that r is regular with valency k whenever |r(x)| = k for all x e X. We say that r is distance-regular whenever for all integers h, i, j (0 < h, i, j < D) and x, y e X with d(x, y) = h, the number
phj := |r(x) n rj(y)|
is independent of x and y. The integers phj are called the intersection numbers of r. From now on assume that r is distance-regular with diameter D > 3. We abbreviate k :=
(0 < i < D). For 0 < i < D let A, denote the matrix in MatX (C) with (x, y)-entry
1 if d(x,y) = i, 0 if d(x,y) = i,
(Ai)xy ={n	x, y e X.
We call A, the i-th distance matrix of r. We call A = A1 the adjacency matrix of r. Observe that A, is real and symmetric for 0 < i < D. Note that A0 = I is the identity matrix in MatX (C). Observe that J2i=0 A, = J, where J is the all-ones matrix in MatX (C). Observe that for 0 < i, j < D,
D
AiAj = Y, phj Ah.	(2.1)
h=0
S. Sumalroj: A diagram associated with the subconstituent algebra of a distance-regular graph 187
For integers h, i, j (0 < h, i, j < D) we have
0	c
P0 j	= A0j,
P0	= A- k-
i-'ij Vij^i-
(2.2) (2.3)
Let M denote the subalgebra of MatX (C) generated by A. By [2, p. 44] the matrices A0, Ai,..., Ad form a basis for M. We call M the Bose-Mesner algebra of r. By [1, p. 59, 64], M has abasis E0, E1,...,ED such that
(i)	Eo = |X|-1 J;
(ii)	ZlLo Ei = I;
(iii)	Et = Ei (0 < i < D);
(iv)	Ei = Ei (0 < i < D);
(v)	EiEj = SijEi (0 < i, j < D).
The matrices E0, E1,..., ED are called the primitive idempotents of r, and E0 is called the trivial idempotent. For 0 < i < D let mi denote the rank of Ei. For 0 < i < D let denote an eigenvalue of A associated with Ei. Let A denote an indeterminate. Define polynomials {ui}:D=0 in C[A] by u0 = 1, u1 = A/k, and
Au, = CjMj_i + aiUi + 6iMi+i (1 < i < D - 1).
By [2, p. 131, 132],
Aj = kjYl uj
i=0
D
Ej = |X|-1mj Y Ui(0j)Ai
(0 < j < D), (0 < j < D).
(2.4)
(2.5)
Since EjEj = SijE, and by (2.4) we have AjE, = kjUj(0i)Ei = E,Aj (0 < i, j < D). By [1, Theorem 3.5] we have the orthogonality relations
D
Y,Ui(0r )Ui(9s)ki = Ars m-1 |X |
i=0 D
YjUi(dr )Uj (0r )mr = ¿j k-^X |
(0 < r, s < D),
(0 < i,j < D).
(2.6)
(2.7)
r=0
We recall the Krein parameters of r. Let o denote the entry-wise multiplication in Matx(C). Note that Ai o Aj = Sij Ai for 0 < i, j < D. So M is closed under o. By [2, p. 48], there exist scalars qj G C such that
D
Ei o Ej = |X|-1 Y qhjEh
0=0
(0 < i,j < D).
(2.8)
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ArsMath. Contemp. 17 (2019) 185-202
We call the qj the Krein parameters of r. By [2, Proposition 4.1.5], these parameters are real and nonnegative for 0 < h, i, j < D.
We recall the dual Bose-Mesner algebra of r. Fix a vertex x g X. For 0 < i < D let E* = E* (x) denote the diagonal matrix in MatX (C) with (y, y)-entry
(E*)y
1 if d(x,y) = i,
y g X.
i)yy \0 if d(x,y) = i, We call E* the i-th dual idempotent of r with respect to x. Observe that
(i)	Ef=0 E* = I;
(ii)	E* = E* (0 < i < D);
(iii)	E* = E* (0 < i < D);
(iv)	E*E* = JijE* (0 < i, j < D).
By construction E*, E*,..., E* are linearly independent. Let M* = M*(x) denote the subalgebra of MatX (C) with basis E*, E*,..., E*. We call M* the dual Bose-Mesner algebra of r with respect to x.
We now recall the dual distance matrices of r. For 0 < i < D let A* = A* (x) denote the diagonal matrix in MatX (C) with (y, y)-entry
(A|)yy = |X |(Ei)
xy
y g X.
(2.9)
We call A* the dual distance matrix of r with respect to x and Ei. By [32, p. 379], the matrices A*, A*,..., A D form a basis for M*. Observe that
(i)	A* = I;
(ii)	Ef=oA* = |X|E*;
(iii)	Af = A* (0 < i < D);
(iv)	A* = A* (0 < i < D);
(v)	A*A* = £D=o qijAh (0 < i, j < D).
From (2.4) and (2.5) we have
D
A* = m-E )E*
¿=0
D
E* = |X|-1kj E uj(0i)A*
¿=0
(0 < j < D), (0 < j < D).
(2.10) (2.11)
S. Sumalroj: A diagram associated with the subconstituent algebra of a distance-regular graph 187
3 The subconstituent algebra T
In this section we study the subconstituent algebra of a distance-regular graph. For the rest of the paper, fix a distance-regular graph r and a vertex x of r. Let T = T(x) denote the subalgebra of MatX (C) generated by M, M*. The algebra T is called the subconstituent algebra (or Terwilliger algebra) [32]. In order to describe T, we consider how M, M* are related. We will use the following notation. For any two subspaces R, S of MatX (C) we define RS = Span{RS | R e R, S e S}. Consider the subspaces M, M * ,MM *,M *M, MM *M, M * MM *,... along with their intersections and sums. To describe the inclusions among the resulting subspaces we draw a diagram; see Figure 1. In this diagram, a line segment that goes upward from U to W means that W contains U.
Consider the diagram in Figure 1. For each subspace U shown in the diagram, we seek an orthogonal basis for U and the dimension of U. Also, for each edge U C W shown in the diagram, we seek an orthogonal basis for the orthogonal complement of U in W along with the dimension of this orthogonal complement. We accomplish these goals for that part of the diagram up to MM * + M *M. Our main results are summarized in Theorems 6.1 and 6.2. Before we get started, we recall a few inner product formulas.
Lemma 3.1 ([11, Lemma 3.1, Lemma 4.1]). For 0 < h, i,j, r,s,t < D,
(i)	(E*AjE*h,E*rAsE*t) = SirSjsShtkhph,
(ii)	{EiA**Eh,ErASEt) = SirSjsShtmhqhjj. The following result is well-known.
Lemma 3.2 ([32, Lemma 3.2]). For 0 < h,i,j < D,
(i)	E*AhE* = 0 if and only ifphhj = 0,
(ii)	EiA*hEj = 0 if and only if qh = 0.
Lemma 3.3 ([29, Lemma 10]). For 0 < h,i, j, r,s,t < D,
D
(AiE*Ah,Ar E*sAt) = £ kepirpiapeht.
4 The subspace M + M *
Our goal in this section is to analyze the inclusion diagram up to M + M *. We begin with the trace of elements in M and M *.
Lemma 4.1. For 0 < i < D,
(i)	tr(Ai) = SoilX|,
(ii)	tr(Ei) = mi,
(iii)	tr(E* ) = ki,
(iv) tr(A*) = SoilX
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Ars Math. Contemp. 17 (2019) 153-183
T
MM * M + M *MM *
MM *M
M*MM*
MM*M M*MM*
MM * + M*M
MM*
M*M
MM* M*M
M + M *
M
M*
M M*
CI
Figure 1: Inclusion diagram.
S. Sumalroj: A diagram associated with the subconstituent algebra of a distance-regular graph 187
Proof. (i): Follows from the definition of Ai.
(ii):	Since Ei is diagonalizable, we have tr(£j) = rank(Ej) = m;.
(iii):	Follows from the definition of E*.
(iv):	By (2.5) and since
D
Eo = |X |-1J = |X |-1 Y A;,
i=o
we have
D
Y(1 - Ui(öo))Aj = 0.
i=o
Since {Ai}D=0 are linearly independent, we obtain ui(60) = 1 for 0 < i < D. By (2.6), (2.10) and (iii), we have
DD
tr(A*) = mi Yuj(Oi)tr(E*) = mi YUj(^i)uj(^)kj = Soi|X|.	□
j=o	j=o
Next we obtain some inner products.
Lemma 4.2. For 0 < i,j < D,
(i)	(Ai,Aj) = dijkilX|,
(ii)	(Ei, Ej) = Sij mi,
(iii)	(E*,E*) = Sijki,
(iv)	(A*,A*) = Sijmi|X|.
Proof. (i): Use (2.1) and Lemma 4.1.
(ii):	By Lemma 4.1 and since EiEj = SijEi.
(iii):	Since E* E* = SijE* (0 < i, j < D) and by Lemma 4.1 (iii).
(iv):	By (2.10) and (iii), we obtain
D	D	D
(A*, A*) = (mi Y uh (^i)E*h,m^Yj ue(@j )E*) = m^j^ uh(di)uh(6j )kh.
h=o	i=o	h=o
By (2.6), we have (A*, A*) = mimjSijm-11X| = Sijm^X|.	□
The algebra M has two bases {Ai}D=o and {Ei}D=o. The algebra M* has two bases
o and {Ei*}D=o. Next we show that thesebases are orthogonal.
Lemma 4.3. Each of the following is an orthogonal basis for M:
{ Ai }D=o,	{Ei}i=o.
Moreover, each of the following is an orthogonal basis for M *:
{A*}D=o,	{Ei*}D=o.
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Proof. By Lemma 4.2 and the comment below it.
□
Recall that A0 = 1 = A*. Next we compute some inner products between M and M*.
Lemma 4.4. For 0 < i, j < D,
(Ai,AQ) = 5i050j |X | ki.
Proof. Observe that (Ai, A*) = (AiAQAo, AoA*Ao). By Lemma 3.3 and (2.2), (2.6) and (2.10), the result follows.	□
The next results describe orthogonal bases for M + M* and M n M*.
Lemma 4.5. The following is an orthogonal basis for M + M *:
Ad ,...,Ai ,1, A1 ...,AD.
Proof. Immediate from Lemmas 4.2 and 4.4.	□
Lemma 4.6.
M n M * = CI and dim(M n M *) = 1. Proof. Observe that I G M n M *. By linear algebra, we have
dim(M n M * ) = dim(M ) + dim(M *) - dim(M + M *). By construction dim(M) = D + 1, dim(M*) = D + 1. By this and Lemma 4.6,
Lemma 4.8. The following statements hold:
(i)	The matrices {Ai}D=1 form an orthogonal basis for the orthogonal complement of M n M * in M.
(ii)	The matrices {A*}D=1 form an orthogonal basis for the orthogonal complement of M n M * in M *.
(iii)	The matrices {Ai}D=1 form an orthogonal basis for the orthogonal complement of M * in M + M *.
(iv)	The matrices {A*}D=1 form an orthogonal basis for the orthogonal complement of M in M + M*.
Proof. Follows from definitions of M, M * along with Lemmas 4.5 and 4.7.	□
Lemma 4.9. Each of the following subspaces has dimension D:
dim(M + M *) = 2D + 1.
Proof. Immediate from Lemma 4.5. Lemma 4.7. We have
□
dim(M n M*) = 1. The result follows.
□
(M n Mn M, (Mn (M + M*),
(M n Mn M*, M ^ n (M + M * ).
Proof. Immediate from Lemma 4.8.
□
S. Sumalroj: A diagram associated with the subconstituent algebra of a distance-regular graph 187
5 The subspace MM * + M *M
Our goal in this section is to analyze the inclusion diagram from M + M * up to MM * + M *M. We begin with a few inner product formulas.
Lemma 5.1. For 0 < i,j, r,s < D,
(i)	(AiA*pA*rAs) = SisSjr\X\kimjui(6j),
(ii)	{AiA*,ArAS) = SirSjs\X\kimj,
(iii)	{A*Aj,A*As) = SirSjs\X\kimj. Proof. (i): Since
{AiA* ,Ar AS) =t'r{A* AiAr As) = ^)yy (Ai)yz (A*r )zz (As )zy
yex zex
and by (2.9), it follows that
{AiA* ,Ar As) = \X\2 £ £ (Ej )Xy (Ai)yz (Er )xz (As)zy
yex zex
= \X\2 £ £ (Ej )xy (Ai O As)yz (Er )zx yex zex
Since Ai o As = SisAi (0 < i, s < D), we get
{AiA* ,A*r As ) = \X\2Sis £ £ (Ej )xy (Ai)yz (Er )zx. yex zex
Since
£ £ (Ej )xy (Ai)yz (Er )zx = \X ^ tl(Ej AîE* ), yex zex
we have
{AiA* ,A** As) = \X\Sis tr(Ej AîE* ) = \X\Sis tr(Er Ej Ai). Since EiEj = SijEi (0 < i,j < D), we obtain
{AiA*,A**As) = \X\SisSjr tr(EjAi) = \X\SisSjr {Ej,Ai).
By (2.5) and Lemma 4.2 (i), we get {Ej, Ai) = mjui(6j)ki. Hence
{AiA* ,Ar As) = SisSjr\X\kimj u (0j ).
(ii): Since A0 = I, we get {AiA*,ArAs) = {AiA*A0,ArAsA0). By (2.10), we obtain
D	D
{AiA* ,Ar As ) = mj m s £uh(0j )£ ue(6s){AiEhAo,Ar E* Ao).
h=0 1=0
From Lemma 3.3 we have
D
{AiEhA0, Ar E* A0) = £ ktPlrPthePt00.
t=0
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Ars Math. Contemp. 17 (2019) 153-183
By (2.2) and (2.3), we obtain
D	D
(AiA* ,ArA*s) = mj msY Uh(Oj ) E ue(es)kop0irp0M
h=0 £=0 D
= ¿ir ki'mj m^Y, Uh(Oj )uh(Os)kh.
h=0
By (2.6), we get
(AiA*, Ar A*) = ¿ir ki'mj msSj8ms | = Sir SjS\X | kimj.
(iii): Since
(A*Aj, A*AS) = tr((A*Aj )i(A*As)) = tr(Aj A*A*AS) = tr(A*A*AsAj)
and
A*A* =	qhrA,
D
„h A*
h
h=0
and by (2.1), we get
D D	D D
(A*Aj, A*As) = E E qhrP^s tr(AhA,) = E E j tr(A,Ah)
h=0£=0 h=0£=0
= E E qhrpjs tr(A£ Ah) = qir j (a* Ah).
h=0£=0 h=0£=0
From Lemma 4.4, we have
D D	D D
EE qhrp£s(A£,Ah) = |X | EE qhr pjs ¿£0Sh0k£ = |X |<Z0rp0sk0 = |X
h=0 £=0 h=0 £=0
By (2.3) and since q0r = Sirmi, we obtain
(A*Aj, A* As) = Sir Sjs|X |kj mi.	□
Next we obtain orthogonal bases for MM * and M *M. Lemma 5.2. The following statements hold:
(i)	The matrices {AiA* | 0 < i, j < D} form an orthogonal basis for MM *.
(ii)	The matrices {A*Ai | 0 < i, j < D} form an orthogonal basis for M *M.
Proof. Immediate from Lemma 5.1.	□
Lemma 5.3. Each of the following subspaces has dimension (D + 1)2: MM*,	M*M.
S. Sumalroj: A diagram associated with the subconstituent algebra of a distance-regular graph 187
Proof. Immediate from Lemma 5.2.	□
Our next goal is to obtain an orthogonal basis for MM * + M *M. Lemma 5.4. We have
D D
MM * + M *M =	SpanjAjA* ,A*Ai} (orthogonal direct sum).
i=0 j=0
Proof. Immediate from Lemma 5.1.	□
Corollary 5.5. We have
DD
dim(MM * + M * M) = ^^ dim(SpanjAi A*, A*Ai}).
i=0 j=0
Proof. Immediate from Lemma 5.4.	□
Definition 5.6. For 0 < i, j < D let Hijj denote the 2 x 2 matrix of inner products for
A . A* A* A ■
Lemma 5.7. For 0 < i,j < D,
1 v,i(0j)
Hij = lXl^i^ (u(öj) Ui(l'j))
ij
Proof. Immediate from Lemma 5.1 and Definition 5.6.	□
Lemma 5.8. For 0 < i,j < D we have
det(Hi,j) = |X|2k2m2(1 - (ui(0j))2).
Proof. Immediate from Lemma 5.7.	□
Corollary 5.9. For 0 < i, j < D, det(Hi,j) = 0 if and only if ui(6j) = ±1.
Proof. Immediate from Lemma 5.8.	□
Lemma 5.10. The following elements are orthogonal for 0 < i, j < D:
A■A*+ A*A■	A-A*— A*A-
AiAj + Aj Ai, AiAj Aj Ai.
Moreover
||AiA* + A*Ai||2 = 2|X | kimj (1 + ui(8j)), ||AiA* - A*Ai||2 = 2|X|kimj(1 - u,i(9j)).
Proof. Immediate from Lemma 5.7.	□
Lemma 5.11. The following statements hold for 0 < i,j < D:
(i) Assume ui(0j) = 1. Then AiA* = A*Ai and this common value is nonzero.
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(ii)	Assume uj (Oj ) = -1. Then AjA* = — A*Aj and this common value is nonzero.
(iii)	Assume uj(Oj) = ±1. Then AjA*, A*Aj are linearly independent.
Proof. (i), (ii): Immediate from Lemma 5.10.
(iii): Immediate from Lemma 5.8.	□
Lemma 5.12. For 0 < i, j < D we give an orthogonal basis for SpanjAjA*, A*Aj} in Table 1.
Table 1: An orthogonal basis for SpanjAjA*, A*Aj}.
Case	Orthogonal basis	Dimension
uj (Oj ) = ±1	AjA*	1
uj (Oj ) = ±1	A-A* + A* A- A-A* A* A-Aj Aj + Aj Aj, Aj Aj Aj Aj	2
Proof. Follows from Definition 5.6 and Lemmas 5.7 and 5.11.	□
Corollary 5.13. The following is an orthogonal basis for MM * + M *M:
{AA* + A*Ai, AiA* - A*Ai | 0 < i,j < D, u^Oj) = ±1}
U {AiA* | 0 < i, j < D, ui(Oj) = ±1}.
Proof. Immediate from Lemmas 5.4 and 5.12.	□
Our next goal is to find the dimension of MM * + M *M. Definition 5.14. Define an integer P as follows:
P = |{(i,j) I 1 < i,j < D,ui(0j) = ±1}|.
Remark 5.15. Recall that u0(Oj) = 1 and Ui(O0) = 1 for 0 < i, j < D. By [2, A.5], the graph r is primitive if and only if ri is connected for 1 < i < D. From Definition 5.14 and [2, Proposition 4.4.7] we have P = 0 if and only if r is primitive.
Lemma 5.16.
dim(MM* + M *M) = 2D2 + 2D + 1 - P. Proof. Immediate from Corollary 5.13 and Definition 5.14.	□
Our next goal is to obtain an orthogonal basis for MM * n M * M. Lemma 5.17.
dim(MM* n M *M) = 2D + 1 + P. Proof. By linear algebra, we have
dim(MM* n M *M) = dim(MM *) + dim(M *M) - dim(MM * + M *M). By Lemmas 5.3 and 5.16, the result follows.	□
S. Sumalroj: A diagram associated with the subconstituent algebra of a distance-regular graph 187
Lemma 5.18. The following is an orthogonal basis for MM * n M * M : {AjA* | 0 < i,j < D, UiiOj) = ±1}.
Proof. Immediate from Lemmas 5.11 and 5.17.	□
We now have orthogonal bases for MM *, M *M, MM * n M *M and MM * + M *M.
The next results establish an orthogonal basis for certain orthogonal complements along
with the dimension for these orthogonal complements.
Lemma 5.19. The matrices {AjA* | 1 < i,j < D,Ui(0j) = ±1} form an orthogonal
basis for the orthogonal complement of M + M * in MM * n M *M.
Proof. Follows from Lemmas 4.5 and 5.18.	□
Lemma 5.20. The following statements hold:
(i)	The matrices {AjA* | 1 < i, j < D,Ui(6j ) = ±1} form an orthogonal basis for the orthogonal complement of MM * n M *M in MM *.
(ii)	The matrices {A*Aj | 1 < i, j < D,Ui(6j ) = ±1} form an orthogonal basis for the orthogonal complement of MM * n M *M in M *M.
Proof. Follows from Lemmas 5.2 and 5.18.	□
Lemma 5.21. The following statements hold:
(i)	The matrices {u(0j)AjA* — A**Ai | 1 < i, j < D,Ui(6j) = ±1} form an orthogonal basis for the orthogonal complement of MM * in MM * + M *M.
(ii)	The matrices {AiA* — ui(6j)A*Ai | 1 < i, j < D,ui(6j) = ±1} form an orthogonal basis for the orthogonal complement of M *M in MM * + M *M.
Proof. (i): By Lemma 5.1, for 0 < i,j,r,s < D
(Ar A** + A** Ar ,Ui(0j )AjA* — A*Aj)
= Ui(9j)(ArA**, AjA*) — (ArA**, A* A) + Uj(ffj)(A**Ar,AjA*) — (A*sAr, A**Aj) = öjr öjs^ | kjmj Uj(0j ) — Sjr Sj^X |kjmj Uj(6j )
+ Sjr Sjs |X (Uj(dj ))2 — Sjr Sjs|X |kj mj = Sjr SjS|X |kjmj ((Uj(6j ))2 — 1).
By similar arguments,
(ArAS — AS Ar ,Uj(0j )AjA* — A*Aj ) = Sjr Sjs|X |kj mj (1 — (u^- ))2)
for 0 < i, j, r, s < D. By Lemma 5.1, for 0 < i, j, r, s < D
(Ar AS ,Uj(0j )AjA* — A* Aj) = Uj(0j )(Ar AS, AjA* ) — (Ar AS, A* Aj)
= Sjr SjS|X |kj mj Uj(0j ) — Sjr SjS|X |kjmj Uj(0j ) = 0.
By Lemma 5.2 and Corollary 5.13, the result follows.
(ii): Similar to the proof of (i).	□
177
Ars Math. Contemp. 17 (2019) 153-183
Lemma 5.22. The following subspace has dimension P:
(M + Mn (MM* n M*M). Proof. Immediate from Definition 5.14 and Lemma 5.19.	□
Lemma 5.23. Each of the following subspaces has dimension D2 — P:
(MM* n M*Mn MM*,	(MM* n M*Mn M*M,
(MM*)L n (MM* + M*M),	(M*Mn (MM* + M*M).
Proof. Immediate from Definition 5.14 and Lemmas 5.20 and 5.21.	□
6 Summary of main results
In Sections 4 and 5 we obtained an orthogonal basis and the dimension for each subspace U in the diagram of Figure 1 up to MM * + M *M. Also, for each edge U C W shown in this part of the diagram of Figure 1, we obtained an orthogonal basis for the orthogonal complement of U in W along with the dimension of this orthogonal complement. The results are summarized in this section.
Theorem 6.1. In each row of Table 2 we describe a subspace U in the diagram of Figure 1. We give an orthogonal basis for U along with the dimension of U.
Table 2: An orthogonal basis for each subspace U in the diagram of Figure 1 along with its dimension.
Subspace U	Orthogonal basis for U	Dimension of U
M n M *	I	1
M	{Ai}E= o	D + 1
M*	{A**}D=o	D + 1
M + M *	[Ad ,...,A1,I,A*,...,A*d }	2D + 1
MM * n M *M	{AjA* | 0 < < D,ui(9j) = ±1}	2D + 1 + P
MM*	{AjA* | 0 < < D}	(D + 1)2
M * M	{A*Aj | 0 < < D}	(D + 1)2
MM * + M *M	(4-4*+ A* 4- 4-4* 4*4-1 {AiAj + Ajaj,ajAj AjAj | o < < D,ui(ej) = ±1} U {AjA* | 0 < < D,Ui (ej) = ±1}	2D2 + 2D + 1 — P
S. Sumalroj: A diagram associated with the subconstituent algebra of a distance-regular graph 187
Theorem 6.2. In each row of Table 3 we describe an edge U Ç W from the diagram of Figure 1. We give an orthogonal basis for the orthogonal complement of U in W along with the dimension of this orthogonal complement.
Table 3: An orthogonal basis for the orthogonal complement of U in W in the diagram of Figure 1 along with the dimension of this orthogonal complement.
U	W	Orthogonal basis for U ^ n W	Dimension of U ^ n W
M n M '	M		D
M n M '	M '		D
M	M + M '	{A'}D=i	D
M '	M + M '	{Ai}D=i	D
M + M '	MM' n M' M	{AjA* | 1 < < ) = ±1}	P
MM ' n M'M	MM '	{AiA* | 1 < < ) = ±1}	D2 - P
MM ' n M'M	M'M	{A*Ai | 1 < < ) = ±1}	D2 - P
MM '	MM ' + M'M	{ui(0j)AiA' - A'Aj | 1 < < D, )= ±1}	D2 - P
M'M	MM ' + M'M	{AiA' - «¿(A;)A'Aj | 1 < i,j < D, uj(0j)= ±1}	D2 - P
7 Open problems
In this section, we give some open problems and suggestions for future research. Earlier in the paper we discussed the diagram of Figure 1. In this discussion we analyzed the diagram up to MM * + M *M. The remaining part of the diagram is not completely understood. We mention what is known. By Lemma 3.1 the subspace M*MM* has an orthogonal basis {E*AjE*h | 0 < h,i, j < D, ph = 0}. Similarly, the subspace MM*M has an orthogonal basis {EiA*Eh | 0 < h, i, j < D, qj = 0}.
Problem 7.1. Find an orthogonal basis for the following subspaces:
(i) MM'M n M'MM',
(ii) MM *M + M 'MM *.
Problem 7.2. In each row of Table 4 we give an edge U Ç W from the diagram of Figure 1. Find an orthogonal basis for the orthogonal complement of U in W for the following cases.
179	Ars Math. Contemp. 17 (2019) 153-183
Table 4: Subspaces U and W from the diagram of Figure 1.
U	W
MM * + M *M	MM * M n M *MM *
MM*M n M*MM*	MM *M
MM*M n M*MM*	M * MM *
MM*M	MM *M + M * MM *
M*MM*	MM *M + M * MM *
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