The Clebsch–Gordan Rule for U ( sl 2 ), the Krawtchouk Algebras and the Hamming Graphs

. Let D ≥ 1 and q ≥ 3 be two integers. Let H ( D ) = H ( D, q ) denote the D -dimensional Hamming graph over a q -element set. Let T ( D ) denote the Terwilliger algebra of H ( D ). Let V ( D ) denote the standard T ( D )-module. Let ω denote a complex scalar. We consider a unital associative algebra K ω defined by generators and relations. The generators are A and B . The relations are A 2 B − 2 ABA + BA 2 = B + ωA , B 2 A − 2 BAB + AB 2 = A + ωB . The algebra K ω is the case of the Askey–Wilson algebras corresponding to the Krawtchouk polynomials. The algebra K ω is isomorphic to U( sl 2 ) when ω 2 ̸ = 1. We view V ( D ) as a K 1 − 2 q -module. We apply the Clebsch–Gordan rule for U( sl 2 ) to decompose V ( D ) into a direct sum of irreducible T ( D )-modules.


Introduction
Throughout this paper, we adopt the following conventions: Fix an integer q ≥ 3. Let C denote the complex number field.An algebra is meant to be a unital associative algebra.An algebra homomorphism is meant to be a unital algebra homomorphism.A subalgebra has the same unit as the parent algebra.In an algebra the commutator [x, y] of two elements x and y is defined as [x, y] = xy − yx.Note that every algebra has a Lie algebra structure with Lie bracket given by the commutator.
Note that the U(sl 2 )-module L n is irreducible for any integer n ≥ 0. Furthermore, the finite-dimensional irreducible U(sl 2 )-modules are classified as follows: H.-W. Huang Lemma 1.3.For any integer n ≥ 0, each (n + 1)-dimensional irreducible U(sl 2 )-module is isomorphic to L n .
It is well known that the universal enveloping algebra of a Lie algebra is a Hopf algebra.For example, see [12,Section 5].
For the rest of this paper, let ω denote a scalar taken from C. Definition 1.6.The Krawtchouk algebra K ω is an algebra over C generated by A and B subject to the relations The algebra K ω is the case of the Askey-Wilson algebra corresponding to the Krawtchouk polynomials [22,Lemma 7.2].Define C to be the following element of K ω : Lemma 1.7.The algebra K ω has a presentation with the generators A, B, C and the relations [C, B] = A + ωB. (1.5) Proof .The relation (1.3) is immediate from the setting of C. Using (1.3), the relations (1.1) and (1.2) can be written as (1.4) and (1.5), respectively.The lemma follows.
Let K ω denote a three-dimensional Lie algebra over C with a basis a, b, c satisfying By Lemma 1.7, the algebra K ω is the universal enveloping algebra of K ω .There is a connection between K ω and U(sl 2 ): The Clebsch-Gordan Rule for U (sl 2 ), the Krawtchouk Algebras and the Hamming Graphs 3 Theorem 1.8.There exists a unique algebra homomorphism ζ : Moreover, if ω 2 = 1, then ζ is an isomorphism and its inverse sends Proof .It is routine to verify the result by using Definition 1.1 and Lemma 1.7.Here we provide another proof by applying [13, Lemmas 2.12 and 2.13].
Let σ : sl 2 (C) → U(sl 2 ) denote the canonical Lie algebra homomorphism that sends e, f , h to E, F , H, respectively.Let τ : K ω → K ω denote the canonical Lie algebra homomorphism that sends a, b, c to A, B, C, respectively.By [13, Lemma 2.12], there exists a unique Lie algebra homomorphism φ : Applying the universal property of K ω to the Lie algebra homomorphism σ • φ, this gives the algebra homomorphism ζ.Suppose that ω 2 = 1.Then φ : K ω → sl 2 (C) is a Lie algebra isomorphism by [13,Lemma 2.13].Applying the universal property of U(sl 2 ) to the Lie algebra homomorphism τ • φ −1 , this gives the inverse of ζ.
In this paper, we relate the above algebraic results to the Hamming graphs.We now recall the definition of Hamming graphs.Let X denote a q-element set and let D be a positive integer.The D-dimensional Hamming graph H(D) = H(D, q) over X is a simple graph whose vertex set is X D and x, y ∈ X D are adjacent if and only if x, y differ in exactly one coordinate.Let ∂ denote the path-length distance function for H(D).Let Mat X D (C) stand for the algebra consisting of the square matrices over C indexed by X D .
The adjacency matrix A(D) ∈ Mat X D (C) of H(D) is the 0-1 matrix such that with respect to x is a diagonal matrix given by for all y ∈ X D .The Terwilliger algebra T (D) of H(D) with respect to x is the subalgebra of Mat X D (C) generated by A(D) and A * (D) [16,17,18].Let V (D) denote the vector space consisting of all column vectors over C indexed by X D .The vector space V (D) has a natural T (D)-module structure and it is called the standard T (D)-module.
In [18], Terwilliger employed the endpoints, dual endpoints, diameters and auxiliary parameters to describe the irreducible modules for the known families of thin Q-polynomial distanceregular graphs with unbounded diameter.In [14], Tanabe gave a recursive construction of irreducible modules for the Doob graphs and his method can be adjusted to the case of H(D).In [5], Go gave a decomposition of the standard module for the hypercube.In [4], Gijswijt, Schrijver and Tanaka described a decomposition of V (D) in terms of the block-diagonalization of T (D).In [11], Levstein, Maldonado and Penazzi applied the representation theory of GL 2 (C) to determine the structure of T (D).In [20], it was shown that V (D) can be viewed as a gl 2 (C)module as well as a sl 2 (C)-module.In [2], Bernard, Crampé, and Vinet found a decomposition of V (D) by generalizing the result on the hypercube.
In this paper, we view V (D) as a K 1− 2 q -module as well as a U(sl 2 )-module in light of Theorem 1.8.Subsequently, we apply Theorem 1.5 to prove the following results: H.-W. Huang Proposition 1.9.Let D be a positive integer.For any integers p and k with 0 ≤ p ≤ D and 0 ≤ k ≤ p 2 , there exists a (p − 2k + 1)-dimensional irreducible T (D)-module L p,k (D) satisfying the following conditions: (i) There exists a basis for L p,k (D) with respect to which the matrices representing A(D) and A * (D) are , respectively.
(ii) There exists a basis for L p,k (D) with respect to which the matrices representing A(D) and A * (D) are Here the parameters , {γ i } p−2k i=1 , {θ i } p−2k i=0 are as follows: for i = 1, 2, . . ., p − 2k, Given a vector space W and a positive integer p, we let Theorem 1.10.Let D be a positive integer.Then the standard The algebra T (D) is a finite-dimensional semisimple algebra.Following from [3, Theorem 25.10], Theorem 1.10 implies the following classification of irreducible T (D)-modules: The paper is organized as follows: In Section 2, we give the preliminaries on the algebra K ω .In Section 3, we prove Proposition 1.9 and Theorems 1.10, 1.11 by using Theorem 1.5.In Appendix A, we give the equivalent statements of Proposition 1.9 and Theorems 1.10, 1.11.

The Krawtchouk algebra 2.1 Finite-dimensional irreducible K ω -modules
Recall the U(sl 2 )-module L n from Lemma 1.2.Recall the algebra homomorphism ζ : K ω → U(sl 2 ) form Theorem 1.8.Each U(sl 2 )-module can be viewed as a K ω -module by pulling back via ζ.We express the U(sl 2 )-module L n as a K ω -module as follows: Lemma 2.1.For any integer n ≥ 0, the matrices representing A, B, C with respect to the basis respectively, where The finite-dimensional irreducible K ω -modules are classified as follows: Theorem 2.2.
(i) If ω = −1, then any finite-dimensional irreducible K ω -module V is of dimension one and there is a scalar µ ∈ C such that Av = µv, Bv = µv for all v ∈ V .
(ii) If ω = 1, then any finite-dimensional irreducible K ω -module V is of dimension one and there is a scalar µ ∈ C such that Av = µv, Bv = −µv for all v ∈ V .
(iii) If ω 2 = 1, then L n is the unique (n + 1)-dimensional irreducible K ω -module up to isomorphism for every integer n ≥ 0.
Proof .(i) Let n ≥ 0 be an integer.Let V denote an (n + 1)-dimensional irreducible K −1module.Since the trace of the left-hand side of (1.1) on V is zero, the elements A and B have the same trace on V .If n = 0 then there exists a scalar µ ∈ C such that Av = Bv = µv for all v ∈ V .
To see Theorem 2.2(i), it remains to assume that n ≥ 1 and we seek a contradiction.Applying the method proposed in [6,7,8], there exists a basis {u i } n i=0 for V with respect to which the matrices representing A and B are of the forms Here {θ i } n i=0 is an arithmetic sequence with common difference −1 and the sequence , where ϕ 0 and ϕ n+1 are interpreted as zero.Solving the above recurrence yields that ϕ i = 0 for all i = 1, 2, . . ., n.Thus the subspace of V spanned by {u i } n i=1 is a nonzero K −1 -module, which is a contradiction to the irreducibility of V .(ii) Using Definition 1.6, it is routine to verify that there exists a unique algebra isomorphism K −1 → K 1 that sends A to A and B to −B.Theorem 2.2(ii) follows from Theorem 2.2(i) and the above isomorphism.
Lemma 2.3.There exists a unique algebra automorphism of Proof .It is routine to verify the lemma by using Lemma 1.7.
Lemma 2.4.Suppose that ω 2 = 1.For any integer n ≥ 0, there exists a basis for the K ω -module L n with respect to which the matrices representing A, B, C are respectively, where Proof .Let L n denote the irreducible K ω -module obtained by twisting the K ω -module L n via the automorphism of K ω given in Lemma 2.3.Recall the basis {v i } n i=0 for L n from Lemma 2.1.Observe that the three matrices described in Lemma 2.4 are the matrices representing A, B, C with respect to the basis {v i } n i=0 for the K ω -module L n .By Theorem 2.2(iii), the K ω -module L n is isomorphic to L n .The lemma follows.
Leonard pairs were introduced in [15,19,21] by P. Terwilliger.Suppose that ω 2 = 1.By Lemmas 2.1 and 2.4, the elements A and B act on the K ω -module L n as a Leonard pair.The result was first stated in [13,Theorem 6.3].

The Krawtchouk algebra as a Hopf algebra
Let H denote an algebra.Recall that H is called a Hopf algebra if there are two algebra homomorphisms ε : H → C, ∆ : H → H ⊗ H and a linear map S : H → H that satisfy the following properties: The Clebsch-Gordan Rule for U (sl 2 ), the Krawtchouk Algebras and the Hamming Graphs 7 Here m : H ⊗ H → H is the multiplication map and ι : C → H is the unit map defined by ι(c) = c1 for all c ∈ C. Note that m is a linear map and ι is an algebra homomorphism.
Suppose that (H1)-(H3) hold.Then the maps ε, ∆, S are called the counit, comultiplication and antipode of H, respectively.Let n be a positive integer.The n-fold comultiplication of H is the algebra homomorphism ∆ n : H → H ⊗(n+1) inductively defined by Here ∆ 0 is interpreted as the identity map of H.We may regard every H ⊗(n+1) -module as an H-module by pulling back via ∆ n .Note that for all i = 1, 2, . . ., n. (2.1) Recall from Section 1 that K ω is the universal enveloping algebra of K ω .Hence K ω is a Hopf algebra.For the reader's convenience, we give a detailed verification for the Hopf algebra structure of K ω .By an algebra antihomomorphism, we mean a unital algebra antihomomorphism.

Lemma 2.5.
(i) There exists a unique algebra homomorphism ε : (ii) There exists a unique algebra homomorphism ∆ : (iii) There exists a unique algebra antihomomorphism S : K ω → K ω given by (iv) The algebra K ω is a Hopf algebra on which the counit, comultiplication and antipode are the above maps ε, ∆, S, respectively.
We can write i ⊗ y where n ≥ 1 is an integer and x i , x i , y i , y Since each of (ι • ε) x (2) i and (ι • ε) y (2) j is a scalar multiple of 1, it follows that By a similar argument, one may show that k = 1.Hence (H2) holds for Evidently, h and h are linear maps.Using Lemma 2.5(ii), (iii) yields that Let x, y be any two elements of K ω and suppose that h 3) and (2.4), one finds that i y (1) .
Using the antihomomorphism property of S, we obtain i y (1) j S y (2) j

S x
(2) i j S y (2) j

S x
(2) i i h(y)S x (2) i .
By a similar argument, one can show that h = ι • ε.Hence (H3) holds for K ω .The result follows.
For the rest of this paper, the notation ∆ will refer to the map from Lemma 2.5(ii) and ∆ n will stand for the corresponding n-fold comultiplication of K ω for every positive integer n.
3 The Clebsch-Gordan rule for U(sl 2 ) and the Hamming graph H(D, q)

Preliminaries on distance-regular graphs
Let Γ denote a finite simple connected graph with vertex set X = ∅.Let ∂ denote the pathlength distance function for Γ.Recall that the diameter D of Γ is defined by For short, we abbreviate Γ(x) = Γ 1 (x).We call Γ distance-regular whenever for all h, i, j ∈ {0, 1, . . ., D} and all x, y ∈ X with ∂(x, y) = h the number |Γ i (x) ∩ Γ j (y)| is independent of x and y.If Γ is distance-regular, the numbers a i , b i , c i for all i = 0, 1, . . ., D defined by for any x, y ∈ X with ∂(x, y) = i are called the intersection numbers of Γ.Here Γ −1 (x) and Γ D+1 (x) are interpreted as the empty set.We now assume that Γ is distance-regular.Let Mat X (C) be the algebra consisting of the complex square matrices indexed by X.For all i = 0, 1, . . ., D the i th distance matrix A i ∈ Mat X (C) is defined by for all x, y ∈ X.The Bose-Mesner algebra M of Γ is the subalgebra of Mat X (C) generated by A i for all i = 0, 1, . . ., D. Note that the adjacency matrix A = A 1 of Γ generates M and the matrices {A i } D i=0 form a basis for M. Since A is real symmetric and dim M = D + 1, it follows that A has D + 1 mutually distinct real eigenvalues θ 0 , θ 1 , . . ., θ D .Set θ 0 = b 0 which is the valency of Γ.There exist unique (the identity matrix), AE i = θ i E i for all i = 0, 1, . . ., D.
The matrices {E i } D i=0 form another basis for M, and E i is called the primitive idempotent of Γ associated with θ i for i = 0, 1, . . ., D.
Observe that M is closed under the Hadamard product .The distance-regular graph Γ is said to be Q-polynomial with respect to the ordering where we interpret b * −1 , c * D+1 as any scalars in C and E −1 , E D+1 as the zero matrix in Mat X (C).
H.-W. Huang We now assume that Γ is Q-polynomial with respect to {E i } D i=0 and fix x ∈ X.For all i = 0, 1, . . ., D let E * i = E * i (x) denote the diagonal matrix in Mat X (C) defined by for all y ∈ X.The matrix E * i is called the i th dual primitive idempotent of Γ with respect to x.The dual Bose-Mesner algebra M * = M * (x) of Γ with respect to x is the subalgebra of Mat X (C) generated by E * i for all i = 0, 1, . . ., D. Since E * i E * j = δ ij E * i the matrices {E * i } D i=0 form a basis for M * .For all i = 0, 1, . . ., D the i th dual distance matrix The matrices {A * i } D i=0 form another basis for M * .Note that A * = A * 1 is called the dual adjacency matrix of Γ with respect to x and A * generates M * [16, Lemma 3.11].
The Terwilliger algebra T of Γ with respect to x is the subalgebra of Mat X (C) generated by M and M * [16, Definition 3.3].The vector space consisting of all complex column vectors indexed by X is a natural T -module and it is called the standard T -module [16, p. 368].Since the algebra T is finite-dimensional, the irreducible T -modules are finite-dimensional.Since the algebra T is closed under the conjugate-transpose map, it follows that T is semisimple.Hence the algebra T is isomorphic to

The adjacency matrix and the dual adjacency matrix of a Hamming graph
Let X be a nonempty set and let n be a positive integer.The notation stands for the n-ary Cartesian product of X.For any x ∈ X n , let x i denote the i th coordinate of x for all i = 1, 2, . . ., n.
Recall that q stands for an integer greater than or equal to 3. For the rest of this paper, we set X = {0, 1, . . ., q − 1} and let D be a positive integer.Definition 3.1.The D-dimensional Hamming graph H(D) = H(D, q) over X has the vertex set X D and x, y ∈ X D are adjacent if and only if x and y differ in exactly one coordinate.
Let ∂ denote the path-length distance function for H(D).Observe that ∂(x, y) = |{i | 1 ≤ i ≤ D, x i = y i }| for any x, y ∈ X D .It is routine to verify that H(D) is a distance-regular graph with diameter D and its intersection numbers are for all i = 0, 1, . . ., D. Let V (D) denote the vector space consisting of the complex column vectors indexed by X D .For convenience we write V = V (1).For any x ∈ X D , let x denote the vector of V (D) with 1 in the x-coordinate and 0 elsewhere.We view any L ∈ Mat X D (C) as the linear map V (D) → V (D) that sends x to Lx for all x ∈ X D .We identify the vector space V (D) with V ⊗D via the linear isomorphism V (D) → V ⊗D given by Let I(D) denote the identity matrix in Mat X D (C) and let A(D) denote the adjacency matrix of H(D).We simply write I = I(1) and A = A(1).Lemma 3.2.Let D ≥ 2 be an integer.Then Proof .Let x ∈ X D be given.Applying x to the right-hand side of (3.3) a straightforward calculation yields that it is equal to The lemma follows.
Using Lemma 3.2, a routine induction yields that A(D) has the eigenvalues Let E i (D) denote the primitive idempotent of H(D) associated with θ i (D) for all i = 0, 1, . . ., D.
Applying Lemma 3.3 yields that for all i = 0, 1, . . ., D, where for all i = 0, 1, . . ., D.Here b * −1 , c * D+1 are interpreted as any scalars in C. Hence H(D) is Q-polynomial with respect to the ordering {E i (D)} D i=0 .Observe that the graph H(D) is vertex-transitive.Without loss of generality, we can consider the dual adjacency matrix A * (D) of H(D) with respect to (0, 0, . . ., 0) ∈ X D .We simply write A * = A * (1).Lemma 3.4.Let D ≥ 2 be an integer.Then Proof .Given y ∈ X D let c y denote the coefficient of ŷ in E 1 (D) • 0⊗D with respect to the basis {x} x∈X D for V (D).By (3.2), we have Suppose that D ≥ 2. Using Lemma 3.3 yields that c y = q −1 c (y 1 ,...,y D−1 ) + q 1−D c y D for all y ∈ X D .Hence for all y ∈ X D .The lemma follows.
Proof .It is straightforward to verify the lemma by using (3.1).
Using Lemmas 3.4 and 3.5, a routine induction yields that 3.3 Proofs of Proposition 1.9 and Theorems 1.10, 1.11 In this subsection, we set Let T (D) denote the Terwilliger algebra of H(D) with respect to (0, 0, . . ., 0) ∈ X D .Definition 3.6.Let V 0 denote the subspace of V consisting of all vectors q−1 i=1 c i î, where c 1 , c 2 , . . ., c q−1 ∈ C with q−1 i=1 c i = 0. Let V 1 denote the subspace of V spanned by 0 and q−1 i=1 î.
Proposition 3.12.For any s ∈ {0, 1} D , the representation r s (D) : Proof .We proceed by induction on D. By Lemma 3.9, the statement is true when D = 1.Suppose that D ≥ 2. For convenience let s = (s 1 , s 2 , . . ., s D−1 ) ∈ {0, 1} D−1 .By Lemma 2.5 and Proposition 3.11, the map r s (D) sends A to Applying the induction hypothesis the above element is equal to By Lemma 3.2, the first term in the right-hand side of the above equation equals 1 q A(D)| Vs(D) .Hence (3.5) holds.By a similar argument, (3.6) holds.The proposition follows.
In light of Proposition 3.12, the T (D)-module V s (D) is a K ω -module for all s ∈ {0, 1} D .Combined with Lemma 3.8, the standard T (D)-module V (D) is a K ω -module.Lemma 3.13.Let p be a positive integer.Then the K ω -module L ⊗p 1 is isomorphic to The Clebsch-Gordan Rule for U (sl 2 ), the Krawtchouk Algebras and the Hamming Graphs 15 Proof .We proceed by induction on p.If p = 1, then there is nothing to prove.Suppose that p ≥ 2. Applying the induction hypothesis yields that the K ω -module L ⊗p 1 is isomorphic to Applying the distributive law of ⊗ over ⊕ the above K ω -module is isomorphic to for all k = 0, 1, . . ., p 2 .Here p−1 k−1 is interpreted as 0 when k = 0.The lemma follows.
Proof of Proposition 1.9.Let p and k be two integers with 0 ≤ p ≤ D and 0 ≤ k ≤ p 2 .Pick any s ∈ {0, 1} D with p = D i=1 s i .By Lemma 3.14, the K ω -module V s (D) contains the irreducible K ω -module L p−2k .Let {v i } p−2k i=0 and {w i } p−2k i=0 denote the two bases for L p−2k described in Lemmas 2.1 and 2.4 with n = p − 2k, respectively.In light of Proposition 3.12, we may view the K ω -submodule L p−2k of V s (D) as an irreducible T (D)-module and denoted by L p,k (D).To see (i) and (ii), one may evaluate the matrices representing A(D) and A * (D) with respect to the bases {v i } p−2k i=0 and {w i } p−2k i=0 for L p,k (D), respectively.The proposition follows.

Theorem 1 . 11 .
Let D be a positive integer.Let P(D) denote the set consisting of all pairs (p, k) of integers with 0 ≤ p ≤ D and 0 ≤ k ≤ p 2 .Let M(D) denote the set of all isomorphism classes of irreducible T (D)-modules.Then there exists a bijection E : P(D) → M(D) given by (p, k) → the isomorphism class of L p,k (D) for all (p, k) ∈ P(D).
irreducible T -modules W End(W ), where the direct sum is over all non-isomorphic irreducible T -modules W . Since the standard T -module is faithful, all irreducible T -modules are contained in the standard T -module up to isomorphism.Let W denote an irreducible T -module.The number min 0≤i≤D {i | E * i W = {0}} is called the endpoint of W .The number min 0≤i≤D {i | E i W = {0}} is called the dual endpoint of W .The support of W is defined as the set {i | 0

H.-W. Huang Proof of Theorem 1 . 11 .p 2 k=0 2 k=0A 2 ≤−d 2 ≤ 1 D − r + 1 D
Since the standard T (D)-module V (D) contains all irreducible T (D)-modules up to isomorphism, the map E is onto.Suppose that there are two pairs (p, k) and (p , k ) in P(D) such that the irreducible T (D)-module L p,k (D) is isomorphic to L p ,k (D).Since they have the same dimension, it follows that p − 2k = p − 2k .(3.7)SinceA * (D) has the same eigenvalues in L p,k (D) and L p ,k (D), it follows from Proposition 1.9 that p − k = p − k .Combined with (3.7), this yields that (p, k) = (p , k ).Therefore, E is one-to-one.Corollary 3.15 ([11, Corollary 3.7]).The algebra T (D) is isomorphic to D p=0 Mat p−2k+1 (C).Moreover, dim T (D) = D+4 4 .Proof .By Theorem 1.11, the algebra T (D) is isomorphic to D p=0 p End(L p,k (D)).Hence dim T (D) is equal to Restatements of Proposition 1.9 and Theorems 1.10, 1.11 Recall the irreducible T (D)-module L p,k (D) from Proposition 1.9.Let r, r * , d, d * denote the endpoint, dual endpoint, diameter, dual diameter of L p,k (D) respectively.It is known from [18, p. 197] that D−d r, r * ≤ D − d.From the results of Section 3.2, we see that r = r * = D + k − p, d = d * = p − 2k.In terms of the parameters r and d, the parameters p and k read as p = 2D − d − 2r, k = D − d − r.Thus we can restate Proposition 1.9 and Theorems 1.10, 1.11 as follows: Proposition A.1.Let D be a positive integer.For any integers d and r with 0 ≤ d ≤ D and Dr ≤ D − d, there exists a (d + 1)-dimensional irreducible T (D)-module M d,r (D) satisfying the following conditions: (i) There exists a basis for M d,r (D) with respect to which the matrices representing A(D) and A * (D) are Gordan Rule for U (sl 2 ), the Krawtchouk Algebras and the Hamming Graphs 17 (ii) There exists a basis for M d,r (D) with respect to which the matrices representing A(D) and A * (D) are {α i } d i=0 , {β i } d−1 i=0 , {γ i } d i=1 , {θ i } d i=0 are as follows:α i = (D − d + i − r)(q − 1) − i − r for i = 0, 1, . . ., d, β i = i + 1 for i = 0, 1, . . ., d − 1, γ i = (q − 1)(d − i + 1) for i = 1, 2, . . ., d, θ i = D(q − 1) − q(i + r) for i = 0, 1, . . ., d.Theorem A.2. Let D be a positive integer.Then the standard T (D)-module V (D) is isomorphic to 2D − d − 2r 2D − d − 2r D − d − r (q − 2) d−D+2r • M d,r (D).We illustrate Theorem A.2 for D = 3 and D = 4: D d r The support of M d,r (D) The multiplicity of M d,r (D) in V (

4 Theorem A. 3 . 2 ≤D − d 2 + 1 •
Let D be a positive integer.Let P(D) denote the set consisting of all pairs (d, r) of integers with 0 ≤ d ≤ D and D−d r ≤ D − d.Let M(D) denote the set of all H.-W. Huang isomorphism classes of irreducible T (D)-modules.Then there exists a bijection P(D) → M(D) given by (d, r) → the isomorphism class of M d,r (D) for all (d, r) ∈ P(D).By Theorem A.3, the structure of an irreducible T (D)-module is determined by its endpoint and its diameter.Also we can restate Corollary 3.15 as follows: Corollary A.4.The algebra T (D) is isomorphic to D d=0 Mat d+1 (C).