An Action of the Tetrahedron Algebra on the Standard Module for the Hamming Graphs and Doob Graphs

Graphs and CombinatoricsDOI 10.1007/s00373-013-1366-0

ORIGINAL PAPER

An Action of the Tetrahedron Algebra on the StandardModule for the Hamming Graphs and Doob Graphs

John Vincent S. Morales · Arlene A. Pascasio

Received: 18 January 2013 / Revised: 8 July 2013© Springer Japan 2013

Abstract We display an action of the tetrahedron algebra � on the standard modulefor any Hamming graph or Doob graph. To do this, we use some results of BrianHartwig concerning tridiagonal pairs of Krawtchouk type.

Keywords Tetrahedron Lie algebra · Distance-regular graph · Tridiagonal pair

Mathematics Subject Classification (2010) 05E30 · 05C50 · 15A04 · 33C45

1 Introduction

In [8] Hartwig and Terwilliger gave a presentation of the three-point sl2 loop algebrausing generators and relations. To obtain this presentation they defined the tetrahedronLie algebra � using generators and relations, and gave an isomorphism from � to thethree-point sl2 loop algebra. Hartwig [7] described the finite-dimensional irreducible�-modules over an algebraically closed field with characteristic 0. He showed thatthese modules are in bijection with a linear algebraic object called a tridiagonal pairof Krawtchouk type. In [12] Ito and Terwilliger introduced a quantum analogue of�, denoted by �q . They showed that �q is related to the Uq(sl2) loop algebra inroughly the same way that � is related to the sl2 loop algebra. They described thefinite-dimensional irreducible �q -modules over an algebraically closed field, underthe assumption that q is not a root of unity. They showed that these modules are inbijection with the tridiagonal pairs of q-geometric type.

J. V. S. Morales (B) · A. A. PascasioMathematics Department, De La Salle University, 2401 Taft Avenue, 1004 Manila, Philippinese-mail: [email protected]

A. A. Pascasioe-mail: [email protected]

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Graphs and Combinatorics

In [14] Ito and Terwilliger exhibited an action of �q on the standard module of anyself-dual distance-regular graph that has classical parameters and base not equal to1. In the present paper, we get an analogous result for any self-dual distance-regulargraph that has classical parameters with base equal to 1. We now describe this familyfrom three points of view. First, this family consists of the Q-polynomial distance-regular graphs whose eigenvalue sequence and dual eigenvalue sequence are bothin arithmetic progression. Second, referring to the notation used in the version ofLeonard’s Theorem by Bannai and Ito [1], this family consists of the distance-regulargraphs of type IIC. Third, by [2, Theorem 6.1.1] and [18], this family consists of theHamming graphs and Doob graphs. In the present paper, for each graph in this familywe display an action of � on the standard module. To obtain this action we use someresults of Hartwig [7] concerning tridiagonal pairs of Krawtchouk type.

Our construction is summarized as follows. It is natural to identify the generatorsof � with the edges of a tetrahedron. For a pair of opposite edges in the tetrahedron,the corresponding �-generators act essentially as the adjacency matrix and dual ad-jacency matrix of the graph. For the remaining edges of the tetrahedron the actionsof the corresponding �-generators are obtained using the split decompositions of thestandard module [14,24]. There are four versions of these decompositions; they arethe (η, μ)-split decompositions where η,μ ∈ {↓,↑}. Using the four split decom-positions, we define six linear transformations B, B∗, K , K ∗, φ, ψ on the standardmodule. Each of B, ψ acts as a scalar multiple of the identity on every summand inthe (↓,↑)-split decomposition. Each of K , φ acts as a scalar multiple of the identityon every summand in the (↓,↓)-split decomposition. B∗ acts as a scalar multiple ofthe identity on every summand in the (↑,↓)-split decomposition. K ∗ acts as a scalarmultiple of the identity on every summand in the (↑,↑)-split decomposition. For thefour remaining edges of the tetrahedron, the corresponding action of the �-generatorscoincides with the action of B − φ, B∗ − φ, K −ψ and K ∗ −ψ. The elements φ,ψcommute with everything in �.

2 Tridiagonal Pairs and Tetrahedron Algebra

We begin this section by recalling the notion of a tridiagonal pair. For more backgroundinformation about tridiagonal pairs, the reader may refer to [9] and [13]. Throughoutthis section F denotes an algebraically closed field with characteristic 0.

Definition 2.1 Let V denote a vector space over F with finite positive dimension. LetA, A∗ denote a pair of linear transformations from V to V satisfying the followingfour conditions:

(i) Each of A, A∗ is diagonalizable on V .(ii) There exists an ordering {Vi }d

i=0 of the eigenspaces of A such that

A∗Vi ⊆ Vi−1 + Vi + Vi+1 (0 ≤ i ≤ d), (1)

where V−1 = Vd+1 = 0.

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(iii) There exists an ordering {V ∗i }δi=0 of the eigenspaces of A∗ such that

AV ∗i ⊆ V ∗

i−1 + V ∗i + V ∗

i+1 (0 ≤ i ≤ δ), (2)

where V ∗−1 = V ∗δ+1 = 0.

(iv) If W is a subspace of V such that both AW ⊆ W and A∗W ⊆ W , then eitherW = 0 or W = V .

We call such pair a tridiagonal pair on V .

We recall a few facts about tridiagonal pairs. Let A, A∗ denote a tridiagonal pairon V as in Definition 2.1. By [9, Lemma 4.5], the integers d and δ from (1) and (2)are equal. We shall call this common value the diameter of A, A∗. An ordering of theeigenspaces of A (resp. A∗) will be called standard whenever it satisfies (1) [resp.(2)]. Suppose {Vi }d

i=0 denote a standard ordering of the eigenspaces of A. Then theordering {Vd−i }d

i=0 is standard and no other ordering is standard. An analogous resultholds for the eigenspaces of A∗. By an eigenvalue sequence (resp. dual eigenvaluesequence) of A (resp. A∗) we mean an ordering of the eigenvalues for A (resp. A∗)for which the corresponding ordering of eigenspaces is standard. Consequently if theordering {θi }d

i=0 is an eigenvalue sequence then so is {θd−i }di=0 . An analogous result

holds for the dual eigenvalues.The next lemma states how one gets a new tridiagonal pair from old.

Lemma 2.2 [13, p. 222] Let V denote a vector space over F with finite positivedimension. Assumeα, α∗, β, β∗ ∈ F where α, α∗ are nonzero. If A, A∗ is a tridiagonalpair on V with diameter d, then so is the pair αA + β I, α∗ A∗ + β∗ I. Moreover, if{θi }d

i=0 (resp.{θ∗

i

}di=0 ) is an eigenvalue sequence (resp. a dual eigenvalue sequence)

then so is {αθi + β}di=0 (resp.

{α∗θ∗

i + β∗}di=0).

Definition 2.3 Let V denote a vector space over F with finite positive dimension.Let A, A∗ denote a tridiagonal pair on V with diameter d. We say that A, A∗ hasKrawtchouk type whenever the sequence {d − 2i}d

i=0 is both an eigenvalue sequenceand a dual eigenvalue sequence.

We now recall the tetrahedron algebra.

Definition 2.4 [8, Definition 1.1] Let S = {1, 2, 3, 4}. Let � denote the Lie algebraover F with generators

{xi j | i, j ∈ S, i �= j}

such that the following relations hold:

(i) For distinct i, j ∈ S,

xi j + x ji = 0.

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(ii) For mutually distinct h, i, j ∈ S,

[xhi , xi j

] = 2xhi + 2xi j .

(iii) For all mutually distinct h, i, j, k ∈ S

[xhi ,

[xhi ,

[xhi , x jk

]]] = 4[xhi , x jk

].

We call � the tetrahedron algebra over F.

We end this section with some results due to Hartwig.

Theorem 2.5 [7, Theorem 1.8] Let V denote a vector space over F with finite positivedimension and let A, A∗ denote a tridiagonal pair on V with diameter d. SupposeA, A∗ has Krawtchouk type. Then there exists a unique �-module structure on Vsuch that the generators x12, x34 act on V as A, A∗, respectively. This �-module isirreducible.

Theorem 2.6 Let V denote a vector space over F with finite positive dimension and letA, A∗ denote a tridiagonal pair on V with diameter d. Suppose A, A∗ has Krawtchouktype. Let VA(λ) (resp. VA∗(λ)) denote the eigenspace of A (resp. A∗) on V correspond-ing to the eigenvalue λ. Then for each generator xrs of � and for each i (0 ≤ i ≤ d),the eigenspace of xrs on V corresponding to the eigenvalue 2i −d is given in the tablebelow:

r s The eigenspace of xrs on V corresponding to the eigenvalue 2i − d1 2 VA(2i − d)3 4 VA∗(2i − d)4 1 [VA∗(d)+ . . .+ VA∗(d − 2i)] ∩ [VA(d − 2i)+ . . .+ VA(−d)]3 2 [VA∗(−d)+ . . .+ VA∗(2i − d)] ∩ [VA(2i − d)+ . . .+ VA(d)]4 2 [VA∗(d)+ . . .+ VA∗(d − 2i)] ∩ [VA(2i − d)+ . . .+ VA(d)]3 1 [VA∗(−d)+ . . .+ VA∗(2i − d)] ∩ [VA(d − 2i)+ . . .+ VA(−d)]

Proof By Theorem 2.5, V is an irreducible �-module and x12 (resp. x34) acts on V asA (resp. A∗). The rest are immediate from [7, Lemma 5.7] and [7, Definition 7.1]. �

3 Distance-Regular Graphs

In this section we review some definitions and basic concepts concerning distance-regular graphs. For more background information, the reader may refer to [1,2] or [6].

For an arbitrary nonempty finite set X , let V := CX denote the C-vector space of

column vectors with coordinates indexed by X. Let MatX(C) denote the algebra ofmatrices over C with rows and columns indexed by X. Observe MatX(C) acts on Vby left multiplication. We call V the standard module. Endow V with the Hermiteaninner product 〈u, v〉 = utv (u, v ∈ V ) where ut denotes the transpose of u and vdenotes the complex conjugate of v. For each x ∈ X , denote by x the vector in V that

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has one in the x-coordinate and zeros in all other coordinates. Observe {y | y ∈ X} isan orthonormal basis for V .

Let = (X, R) denote a finite, undirected, connected graph without loops ormultiple edges, with vertex set X , edge set R, path-length distance function ∂ anddiameter D := max{∂(x, y) | x, y ∈ X}. We say is distance-regular whenever forall integers h, i, j (0 ≤ h, i, j ≤ D) and for all x, y ∈ X with ∂(x, y) = h, thenumber

phi j := |{z ∈ X |∂(x, z) = i, ∂(y, z) = j}|

is independent of x and y. The integers phi j are called the intersection numbers for .

We abbreviate ai := pi1i (0 ≤ i ≤ D), bi := pi

1i+1(0 ≤ i ≤ D − 1), ci := pi1i−1(1 ≤

i ≤ D), ki := p0i i (0 ≤ i ≤ D), and for convenience we set c0 := 0 and bD := 0.

For the rest of the paper denotes a distance-regular graph with diameter D ≥ 1.We now recall the Bose–Mesner algebra of . For each integer i (0 ≤ i ≤ D), let

Ai denote the matrix in MatX(C) with (x, y) entry

(Ai )xy ={

1, ∂(x, y) = i;0, ∂(x, y) �= i.

(x, y ∈ X).

The matrices A0, A1, . . . , AD are called the distance matrices of .We call A1 theadjacency matrix of . Observe that

A0 = I,

A0 + A1 + · · · + AD = J,

Ati = Ai (0 ≤ i ≤ D),

Ai A j =D∑

h=0

phi j Ah (0 ≤ i, j ≤ D),

where I and J denote the identity and the all ones matrices in MatX(C), respectively.Therefore {Ai }D

i=0 forms a basis for a commutative subalgebra M of MatX(C), calledthe Bose–Mesner algebra of .

By [1, p. 190], A1 generates M.By [1, pp. 59, 64], M has a second basis E0, E1, . . . ,

ED such that

E0 = |X |−1 J,E0 + E1 + · · · + ED = I,

Ei = Ei (0 ≤ i ≤ D),Et

i = Ei (0 ≤ i ≤ D),Ei E j = δi j Ei (0 ≤ i, j ≤ D).

The matrices E0, E1, . . . , ED are called the primitive idempotents of .Since {Ei }D

i=0 forms a basis for M , there exist complex scalars θ0, θ1, . . . , θD such

that A1 = ∑Di=0 θi Ei . Observe that

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A1 Ei = Ei A1 = θi Ei (0 ≤ i ≤ D).

The scalars {θi }Di=0 are real by [1, p. 97], and are mutually distinct since A1 generates

M. We call θi the eigenvalue of associated with Ei (0 ≤ i ≤ D). Observe that Vdecomposes as

V =D∑

i=0

Ei V (orthogonal direct sum).

For 0 ≤ i ≤ D, the space Ei V is the eigenspace of A1 associated with θi .

We recall the Q-polynomial property. Let ◦ denote entry-wise multiplication inMatX(C). Then Ai ◦ A j = δi j Ai (0 ≤ i, j ≤ D). Therefore M is closed under ◦.Thus there exist complex scalars qh

i j (0 ≤ h, i, j ≤ D) such that

Ei ◦ E j = |X |−1D∑

h=0

qhi j Eh (0 ≤ i, j ≤ D).

By [2, p.170], the scalars qhi j are real and nonnegative for 0 ≤ h, i, j ≤ D. We say

is Q-polynomial (with respect to a given ordering E0, E1, . . . , ED) whenever for alldistinct integers h, j (0 ≤ h, j ≤ D), qh

1 j = 0 if and only if |h − j | �= 1.For the rest of this section, we assume that is Q-polynomial with respect to

the ordering E0, E1, . . . , ED of the primitive idempotents. We recall the dual Bose–Mesner algebra of . Fix a vertex x ∈ X. For each integer i (0 ≤ i ≤ D), letE∗

i := E∗i (x) denote the diagonal matrix in MatX(C) with (y, y) entry

(E∗i )yy = (Ai )xy (y ∈ X).

The matrices E∗0 , E∗

1 , . . . , E∗D are called the dual idempotents of with respect to x .

Observe that

E∗0 + E∗

1 + · · · + E∗D = I,

E∗i = E∗

i (0 ≤ i ≤ D),

E∗i

t = E∗i (0 ≤ i ≤ D),

E∗i E∗

j = δi j E∗i (0 ≤ i, j ≤ D).

Therefore {E∗i }D

i=0 forms a basis for a commutative subalgebra M∗ = M∗(x) ofMatX(C), called the dual Bose–Mesner algebra of with respect to x . For conve-nience, we set E∗−1 = 0, and E∗

D+1 = 0.For each integer i (0 ≤ i ≤ D), let A∗

i = A∗i (x) denote the diagonal matrix in

MatX(C) with (y, y) entry

(A∗i )yy = |X |(Ei )xy (y ∈ X).

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By [20, p. 379], the matrices A∗0, A∗

1, . . . , A∗D form a second basis for M∗. More-

over,

A∗0 = I,

A∗0 + A∗

1 + · · · + A∗D = |X |E∗

0 ,

A∗i

t = A∗i (0 ≤ i ≤ D),

A∗i A∗

j =D∑

h=0

qhi j A∗

h (0 ≤ i, j ≤ D).

The matrices A∗0, A∗

1, . . . , A∗D are called the dual distance matrices of with respect

to x . We call A∗1 the dual adjacency matrix of with respect to x . The matrix A∗

1generates M∗ by [20, Lemma 3.11].

We recall the dual eigenvalues of. Since {E∗i }D

i=0 forms a basis for M∗, there existcomplex scalars θ∗

0 , θ∗1 , . . . , θ

∗D such that

A∗1 =

D∑

i=0

θ∗i E∗

i .

Observe that

A∗1 E∗

i = E∗i A∗

1 = θ∗i E∗

i (0 ≤ i ≤ D).

The scalars θ∗0 , θ

∗1 , . . . , θ

∗D are real by [20, Lemma 3.11], and are mutually distinct

since A∗1 generates M∗. We call θ∗

i the dual eigenvalue of associated with E∗i (0 ≤

i ≤ D).Observe that E∗

i V = span{y | y ∈ X, ∂(x, y) = i} (0 ≤ i ≤ D). Using this andthe fact that {y | y ∈ X} is an orthonormal basis for V , we find V decomposes as

V =D∑

i=0

E∗i V (orthogonal direct sum).

For 0 ≤ i ≤ D the space E∗i V is the eigenspace of A∗

1 associated with θ∗i . We call

E∗i V the i th subconstituent of with respect to x .The subalgebra T = T (x) of MatX(C) generated by M and M∗ is called the

Terwilliger algebra (or subconstituent algebra) of with respect to x . It followsthat T is generated by A1 and A∗

1.Moreover, T has finite positive dimension. ObserveT is closed under the conjugate transpose map, so T is semi-simple. By[20, Lemma 3.2],

A1 E∗i V ⊆ E∗

i−1V + E∗i V + E∗

i+1V (0 ≤ i ≤ D),

A∗1 Ei V ⊆ Ei−1V + Ei V + Ei+1V (0 ≤ i ≤ D).

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We recall four direct sum decompositions of V , called the (η, μ)-split decomposi-tions, introduced by Ito and Terwilliger in [14, Definition 10.1]. For −1 ≤ i, j ≤ Dwe define

V ↓↓i, j = (E∗

0 V + · · · + E∗i V ) ∩ (E0V + · · · + E j V ),

V ↑↓i, j = (E∗

DV + · · · + E∗D−i V ) ∩ (E0V + · · · + E j V ),

V ↓↑i, j = (E∗

0 V + · · · + E∗i V ) ∩ (EDV + · · · + ED− j V ),

V ↑↑i, j = (E∗

DV + · · · + E∗D−i V ) ∩ (EDV + · · · + ED− j V ).

In each of the four cases we interpret the right-hand side to be 0 if i = −1 or j = −1.Observe that for η,μ ∈ {↓,↑} and for 0 ≤ i, j ≤ D

V η,μi−1, j + V η,μ

i, j−1 ⊆ V η,μi, j . (3)

We define V η,μi, j to be the orthogonal complement of the left hand side in the right hand

side of inclusion (3), that is,

V η,μi, j = (V η,μ

i−1, j + V η,μi, j−1)

⊥ ∩ V η,μi, j .

By [14, Lemma 10.3], for η,μ ∈ {↓,↑} the standard module V decomposes as

V =D∑

i=0

D∑

j=0

V η,μi, j (direct sum),

called the (η, μ)-split decomposition.

4 Irreducible T -Modules

Throughout this section, we adopt the following notation and assumption.

Assumption 4.1 Let = (X, R) denote a distance-regular graph with diameter D.Assume is Q-polynomial with respect to the ordering E0, E1, . . . , ED of the prim-itive idempotents. Let A1 denote the adjacency matrix of . Fix x ∈ X and writeA∗

1 = A∗1(x), E∗

i = E∗i (x) (0 ≤ i ≤ D), T = T (x), and V = C

X . For 0 ≤ i ≤ D,let θi (resp. θ∗

i ) denote the eigenvalue (resp. dual eigenvalue) of associated with Ei

(resp. E∗i ).

By a T -module we mean a subspace W of V such that BW ⊆ W for all B ∈T . A T -module W is said to be irreducible whenever W �= 0 and W contains noT -modules other than 0 and W. Since T is closed under the conjugate-transpose map,for any T -module W the subspace

W ⊥ := {v ∈ V |〈v,w〉 = 0 for all w ∈ W }

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is also a T -module. Moreover, for any T -module U ⊆ W , there exists a uniqueT -module U ′ ⊆ W such that

W = U + U ′ (orthogonal direct sum).

It follows that any T -module is an orthogonal direct sum of irreducible T -modules.In particular, the standard module V can be decomposed as an orthogonal direct sumof irreducible T -modules.

Let W denote an irreducible T -module. Set

ρ := min{i | 0 ≤ i ≤ D, E∗i W �= 0},

d := |{i | 0 ≤ i ≤ D, E∗i W �= 0}| − 1.

We refer to the scalars ρ and d as the endpoint and diameter of W , respectively. Set

τ := min{i | 0 ≤ i ≤ D, Ei W �= 0},d∗ := |{i | 0 ≤ i ≤ D, Ei W �= 0}| − 1.

We refer to τ and d∗ as the dual endpoint and dual diameter of W , respectively. Thediameter d is equal to the dual diameter d∗ (see [17, Corollary 3.3]).

We enumerate some results that will be useful later.

Lemma 4.2 [24, Lemma 3.1] With Assumption 4.1, let W denote an irreducibleT -module with endpoint ρ, dual endpoint τ , and diameter d. The following hold:

(i) For 0 ≤ i ≤ D, we have E∗i W �= 0 if and only if ρ ≤ i ≤ ρ + d; and

W =d∑

h=0

E∗ρ+h W (orthogonal direct sum).

(ii) For 0 ≤ i ≤ D, we have Ei W �= 0 if and only if τ ≤ i ≤ τ + d; and

W =d∑

h=0

Eτ+h W (orthogonal direct sum).

Lemma 4.3 [24, Lemma 3.2] With Assumption 4.1, let W denote an irreducibleT -module with endpoint ρ, dual endpoint τ , and diameter d. The following hold:

(i) E∗i W = E∗

i V ∩ W (0 ≤ i ≤ D).(ii) A1 E∗

ρ+i W ⊆ E∗ρ+i−1W + E∗

ρ+i W + E∗ρ+i+1W (0 ≤ i ≤ d).

(iii) Ei W = Ei V ∩ W (0 ≤ i ≤ D).(iv) A∗

1 Eτ+i W ⊆ Eτ+i−1W + Eτ+i W + Eτ+i+1W (0 ≤ i ≤ d).

It turns out that Q-polynomial distance-regular graphs give an example of a tridi-agonal pair.

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Lemma 4.4 [9, Example 1.4] With Assumption 4.1, let W denote an irreducible T -module with endpoint ρ, dual endpoint τ , and diameter d. Then A1, A∗

1 form a tridi-agonal pair on W with eigenvalue sequence {θτ+i }d

i=0 and dual eigenvalue sequence{θ∗ρ+i

}d

i=0.

We end this section by recalling some results found in [15].

Lemma 4.5 [15, Lemma 5.4] With Assumption 4.1, let W denote an irreducible T -module with endpoint ρ, dual endpoint τ , and diameter d. Then for η,μ ∈ {↓,↑}, Wdecomposes as

W =d∑

i=0

W η,μi (direct sum)

where for 0 ≤ i ≤ d,

W ↓↓i = (E∗

ρW + · · · + E∗ρ+i W ) ∩ (EτW + · · · + Eτ+d−i W ),

W ↑↓i = (E∗

ρ+d−i W + · · · + E∗ρ+d W ) ∩ (EτW + · · · + Eτ+d−i W ),

W ↓↑i = (E∗

ρW + · · · + E∗ρ+i W ) ∩ (Eτ+i W + · · · + Eτ+d W ),

W ↑↑i = (E∗

ρ+d−i W + · · · + E∗ρ+d W ) ∩ (Eτ+i W + · · · + Eτ+d W ).

Lemma 4.6 [15, Lemma 6.6] With Assumption 4.1, let W denote an irreducibleT -module with endpoint ρ, dual endpoint τ, and diameter d. Then the followinghold for 0 ≤ h ≤ d and 0 ≤ i, j ≤ D:

(i) W ↓↓h ⊆ V ↓↓

i, j if and only if i = ρ + h and j = τ + d − h,

(ii) W ↑↓h ⊆ V ↑↓

i, j if and only if i = D − ρ − d + h and j = τ + d − h,

(iii) W ↓↑h ⊆ V ↓↑

i, j if and only if i = ρ + h and j = D − τ − h,

(iv) W ↑↑h ⊆ V ↑↑

i, j if and only if i = D − ρ − d + h and j = D − τ − h,

where W ↓↓h ,W ↑↓

h ,W ↓↑h and W ↑↑

h are as in Lemma 4.5.

5 The Main Result

Assume that = (X, R) and ′ = (X ′, R′) are distance-regular graphs. By theCartesian product of and ′, we mean the graph with vertex set

X × X ′ = { (a, b) | a ∈ X and b ∈ X ′}

such that two vertices (a, b) and (c, d) are adjacent whenever either (i) a is adjacentto c in and b = d or (ii) b is adjacent to d in ′ and a = c. We now mention somegraphs that are formed by taking Cartesian products.

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We say a graph is complete whenever any two distinct vertices in the graph areadjacent. Pick an integer n ≥ 3. The complete graph on n vertices is denoted byKn . Let D denote a positive integer. By the Hamming graph H(D, n) we mean theCartesian product of D copies of Kn (see [2, p. 27] for more information). By [2, p.261], H(D, n) is distance-regular with diameter D, and intersection numbers

ai = i(n − 2), bi = (D − i)(n − 1), ci = i, (0 ≤ i ≤ D).

Its eigenvalues are {n(D − i)− D | 0 ≤ i ≤ D}. The graph H(D, n) is Q-polynomialwith respect to the ordering E0, E1, . . . , ED where Ei is the primitive idempotentcorresponding to the eigenvalue n(D − i)− D.

The Shrikhande graph S has vertex set X consisting of the codewords 000000,110000, 010111, 011011 and those obtained by a cyclic permutation of the six entries.The edge set is

R = {ab | a, b ∈ X and a, b differ in exactly two coordinates}.

The graph S is distance-regular with diameter 2, and its intersection numbers coincidewith those of the Hamming graph H(2, 4).

Let m and n denote a positive and a nonnegative integer, respectively. The Doobgraph D(m, n) is the Cartesian product of m copies of the Shrikhande graph S and ncopies of the complete graph K4. The graph D(m, n) was introduced in [3] by Doob.In [4] Egawa proved that the Doob graph D(m, n) is distance-regular with diameter2m + n and has the same intersection numbers as the Hamming graph H(2m + n, 4).Its eigenvalues are {3(2m + n) − 4i | 0 ≤ i ≤ 2m + n}. The graph D(m, n) isQ-polynomial with respect to the ordering E0, E1, . . . , E2m+n where Ei is the prim-itive idempotent corresponding to the eigenvalue 3(2m + n)− 4i.

We mention some studies involving the Hamming graphs and Doob graphs that arerelevant to the present paper. In [4] Egawa classified all the distance-regular graphswhich have the same parameters as the Hamming graph H(D, n). In [20–22], Ter-williger determined the thin irreducible T -modules for the Q-polynomial distance-regular graphs and demonstrated which irreducible T -modules occur in the Hamminggraphs. In [5] Go gave a comprehensive description of the irreducible T -modules forthe hypercube H(D, 2). In [19] Tanabe determined the irreducible T -modules for theDoob graphs. His work motivated a wide ranging investigation that ultimately led tothe discovery of the tetrahedron algebra [9–11,23]. In this paper, we investigate theHamming graphs and Doob graphs from the point of view of the tetrahedron algebra(see [16] for related work).

For the rest of this section, we make the following assumption.

Assumption 5.1 Let = (X, R) denote a distance-regular graph with diameter D.Assume is Q-polynomial with respect to the ordering E0, E1, . . . , ED of the prim-itive idempotents. Suppose further that has eigenvalues and dual eigenvalues inarithmetic progression. Then there exist nonzero scalars b, b∗ such that θi = θ0 − biand θ∗

i = θ∗0 − b∗i for all i (0 ≤ i ≤ D). We fix the scalars b, b∗. Let V denote

the standard module and let A1 denote the adjacency matrix of . For a fixed vertex

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x , write A∗1 = A∗

1(x), T = T (x) and E∗i = E∗

i (x) (0 ≤ i ≤ D). Let W denote anirreducible T -module with endpoint ρ, dual endpoint τ and diameter d.

We now define linear transformations A, A∗,�,�∗ on V .

Definition 5.2 With Assumption 5.1, define unique linear transformations A, A∗ onV such that

A = 2b−1 A1,

A∗ = 2b∗−1 A∗1.

Definition 5.3 With Assumption 5.1, define unique linear transformations �,�∗ onV such that for every irreducible T -module W , the action on W of each pair in thetable below coincide:

For the rest of the paper we let A, A∗ be as in Definition 5.2 and we let �,�∗ beas in Definition 5.3.

Lemma 5.4 With Assumption 5.1, the following hold for 0 ≤ i ≤ d:

(i) E∗ρ+i W is the eigenspace of A∗ −�∗ on W with eigenvalue d − 2i.

(ii) Eτ+i W is the eigenspace of A −� on W with eigenvalue d − 2i.

Proof Routine. �Theorem 5.5 With Assumption 5.1, the pair A −�, A∗ −�∗ is a tridiagonal pair onW with diameter d. Moreover, A −�, A∗ −�∗ has Krawtchouk type.

Proof Immediate from Lemma 2.2, Lemma 4.4, Definitions 5.2 and 5.3. �Corollary 5.6 With Assumption 5.1, there exists a �-module structure on W for whichthe generators x12, x34 act on W as A − �, A∗ − �∗, respectively. This �-modulestructure is irreducible. Moreover, for each generator xrs of � and for each i (0 ≤i ≤ d), the eigenspace of xrs on W corresponding to the eigenvalue 2i − d is givenin the table below:

r s The eigenspace of xrs on W corresponding to the eigenvalue 2i − d1 2 Eτ+d−i W3 4 E∗

ρ+d−i W

4 1 W ↓↑i

3 2 W ↑↓i

4 2 W ↓↓i

3 1 W ↑↑i

where W ↓↑i ,W ↑↓

i ,W ↓↓i and W ↑↑

i are as in Lemma 4.5.

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Proof The first two statements follow immediately from Theorems 2.5 and 5.5. Toprove the assertions in the table, apply Theorem 2.6 to the tridiagonal pair A−�, A∗−�∗ on W and simplify using Lemmas 4.5 and 5.4. �

Definition 5.7 With Assumption 5.1, define unique linear transformations B, B∗, K ,K ∗, φ and ψ on V such that the following hold for 0 ≤ i, j ≤ D :

The transformation is 0 on

B − (i − j)I V ↓↑i, j

B∗ − ( j − i)I V ↑↓i, j

K − (i − j)I V ↓↓i, j

K ∗ − ( j − i)I V ↑↑i, j

φ − (i + j − D)I V ↓↓i, j

ψ − (i + j − D)I V ↓↑i, j

For the rest of the paper we let B, B∗, K , K ∗, φ and ψ be as in Definition 5.7.

Lemma 5.8 With Assumption 5.1, the following hold:

(i) The transformation φ − (ρ + τ + d − D)I is 0 on W.(ii) The transformation ψ − (ρ − τ)I is 0 on W.

Proof We prove (i). For 0 ≤ h ≤ d, let W ↓↓h be as in Lemma 4.5. By Lemma

4.6(i), W ↓↓h ⊆ V ↓↓

i, j where i = ρ + h and j = τ + d − h. By Definition 5.7, the

transformation φ − (ρ + τ + d − D)I is 0 on W ↓↓h . Since W decomposes as a direct

sum W = ∑dh=0 W ↓↓

h in view of Lemma 4.5, statement (i) holds. The proof of (ii) issimilar. �

Lemma 5.9 With Assumption 5.1, the following hold for 0 ≤ i ≤ d:

(i) The transformation (B − φ)− (2i − d)I is 0 on W ↓↑i .

(ii) The transformation (B∗ − φ)− (d − 2i)I is 0 on W ↑↓i .

(iii) The transformation (K − ψ)− (2i − d)I is 0 on W ↓↓i .

(iv) The transformation (K ∗ − ψ)− (d − 2i)I is 0 on W ↑↑i .

Proof We prove (i). For 0 ≤ i ≤ d, let W ↓↑i be as in Lemma 4.5. By Lemma 4.6,

W ↓↑i ⊆ V ↓↑

k, j where k = ρ+i and j = D−τ−i.By Definition 5.7, the transformation

B − (ρ − D + τ + 2i)I is 0 on W ↓↑i . By Lemmas 4.5 and 5.8, the transformation

φ − (ρ + τ + d − D)I is 0 on W ↓↑i . Let M1 and M2 denote B − (ρ − D + τ + 2i)I

and φ− (ρ + τ + d − D)I , respectively. Statement (i) holds since the transformationM1 − M2 is 0 on W ↓↑

i . We prove (ii), (iii) and (iv) similarly. �

We now prove the main theorem.

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Theorem 5.10 With Assumption 5.1, on V

The action of coincides with the action ofx12 A −�

x34 A∗ −�∗x41 B − φ

x23 B∗ − φ

x42 K − ψ

x13 K ∗ − ψ

Proof Since V is a direct sum of irreducible T -modules, it suffices to show thatthe actions on W coincide. By Corollary 5.6, the generators x12, x34 act on W asA −�, A∗ −�∗, respectively. We now show that the actions on W of x41 and B − φ

coincide. By Lemma 4.5, W = ∑di=0 W ↓↑

i . Pick an integer i (0 ≤ i ≤ d). By Lemma

5.9 and Corollary 5.6, we find that x41 and B − φ both act on W ↓↑i as (2i − d)I. The

remaining statements are proved similarly. �Lemma 5.11 With Assumption 5.1, each of

A, A∗, B, B∗, K , K ∗,�,�∗, φ, ψ (4)

is contained in T .

Proof Since T is semi-simple it suffices to show that each transformation in (4) leavesinvariant every irreducible T -module W (see proof of [14, Lemma 12.1] for details).The result holds for A, A∗ by Definition 5.2, for�,�∗ by Definition 5.3, for φ,ψ byLemma 5.8, and for the rest by Lemmas 4.5 and 5.9. �Theorem 5.12 With Assumption 5.1, let ϑ : � → MatX(C) denote the Lie algebrahomomorphism which maps every generator in � to an element of MatX(C) as inTheorem 5.10. Then the following hold:

(i) The image of � under ϑ is contained in T .(ii) �,�∗, φ, ψ are in the center of T .

(iii) T is generated by ϑ(�) together with �,�∗.

Proof Statement (i) follows from Lemma 5.11. Statement (ii) follows from Definition5.3 and Lemma 5.8. Statement (iii) follows from the fact that T is generated by A andA∗. �Acknowledgments Initial work on the tetrahedron algebra and the Hamming graphs was done by thesecond author and Paul Terwilliger in 2005. We offered him co-authorship of this paper but he declined.We highly appreciate his valuable contribution to this project. We also thank the anonymous referee for theconstructive comments that improved the quality of our manuscript. Finally, we acknowledge the UniversityResearch Coordination Office of De La Salle University for its support.

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An Action of the Tetrahedron Algebra on the Standard Module for the Hamming Graphs and Doob Graphs