Irreducible T-module Research Articles

Certain graphs with exactly one irreducible T-module with endpoint 1, which is thin

Certain graphs with exactly one irreducible T-module with endpoint 1, which is thin

On bipartite graphs with exactly one irreducible [formula omitted]-module with endpoint 1, which is thin

Let Γ denote a finite, simple, connected and bipartite graph. Fix a vertex x of Γ and let T=T(x) denote the Terwilliger algebra of Γ with respect to x. Assume that x is a distance-regularized vertex, which is not a leaf. We consider the property that Γ has, up to isomorphism, a unique irreducible T-module with endpoint 1, and that this T-module is thin. The main result of the paper is a combinatorial characterization of this property.

On the Terwilliger algebra of certain family of bipartite distance-regular graphs with Δ_2 = 0

Let Γ denote a bipartite distance-regular graph with diameter D ≥ 4 and valency k ≥ 3. Let X denote the vertex set of Γ, and let Ai (0 ≤ i ≤ D) denote the distance matrices of Γ. We abbreviate A := A1. For x ∈ X and for 0 ≤ i ≤ D, let Γi(x) denote the set of vertices in X that are distance i from vertex x. Fix x ∈ X and let T = T(x) denote the subalgebra of MatX(ℂ) generated by A, E0*, E1*, …, ED*, where for 0 ≤ i ≤ D, Ei* represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. By the endpoint of an irreducible T-module W we mean min{i ∣ Ei*W ≠ 0}. In this paper we assume Γ has the property that for 2 ≤ i ≤ D − 1, there exist complex scalars αi, βi such that for all y, z ∈ X with ∂(x, y) = 2, ∂(x, z) = i, ∂(y, z) = i, we have αi + βi|Γ1(x) ∩ Γ1(y) ∩ Γi − 1(z)| = |Γi − 1(x) ∩ Γi − 1(y) ∩ Γ1(z)|. We study the structure of irreducible T-modules of endpoint 2. Let W denote an irreducible T-module with endpoint 2, and let v denote a nonzero vector in E2*W. We show that W = span({Ei*Ai − 2E2*v ∣ 2 ≤ i ≤ D} ∪ {Ei*Ai + 2E2*v ∣ 2 ≤ i ≤ D − 2}). It turns out that, except for a particular family of bipartite distance-regular graphs with D = 5, this result is already known in the literature. Assume now that Γ is a member of this particular family of graphs. We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2 and it is not thin. We give a basis for this T-module.

On the Terwilliger algebra of distance-biregular graphs

Let Γ denote a distance-biregular graph with vertex set X. Fix x∈X and let T=T(x) denote the Terwilliger algebra of Γ with respect to x. In this paper we consider irreducible T-modules with endpoint 1. We show that there are no such modules if and only if Γ is the complete bipartite graph K1,n(n≥1) and x is a vertex of Γ with valency 1. If the valency of x is at least 2 then we show that up to isomorphism there is a unique irreducible T-module of endpoint 1, and this module is thin.

A Group Commutator Involving the Last Distance Matrix and Dual Distance Matrix of a Q-Polynomial Distance-Regular Graph: The Hamming Graph Case

Let $$\varGamma $$ denote the Hamming graph H(D, r) with $$r \ge 3$$ . Consider the distance matrices $$\{A_i\}_{i=0}^{D}$$ of $$\varGamma $$ . Fix a vertex x of $$\varGamma $$ , and consider the dual distance matrices $$\{A_i^{*}\}_{i=0}^{D}$$ of $$\varGamma $$ with respect to x. We investigate the group commutator $$A_{D}^{-1}A_{D}^{*-1}A_{D}A_{D}^{*}$$ . We show that this matrix is diagonalizable. We compute its eigenvalues and their eigenspaces. Let T denote the subconstituent algebra of $$\varGamma $$ with respect to x. We describe the action of $$A_{D}^{-1}A_{D}^{*-1}A_{D}A_{D}^{*}$$ on each irreducible T-module.

On bipartite distance-regular graphs with exactly one non-thin [formula omitted]-module with endpoint two

Let Γ denote a bipartite distance-regular graph with diameter D≥4 and valency k≥3. Let X denote the vertex set of Γ, and let A denote the adjacency matrix of Γ. For x∈X and for 0≤i≤D, let Γi(x) denote the set of vertices in X that are distance i from vertex x. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222.For x∈X let T=T(x) denote the subalgebra of MatX(C) generated by A,E0∗,E1∗,…,ED∗, where for 0≤i≤D, Ei∗ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. An irreducible T-module W is said to be thin whenever dim(Ei∗W)≤1 for 0≤i≤D. By the endpoint of an irreducible T-module W we mean min{i|Ei∗W≠0}.Fix x∈X and assume that Γ has, up to isomorphism, exactly one irreducible T-module W with endpoint 2, and that W is non-thin with dim(E2∗W)=1, dim(ED−1∗W)≤1 and dim(Ei∗W)≤2 for 3≤i≤D. We prove that for 2≤i≤D, there exist complex scalars αi,βi such that |Γi−1(x)∩Γi−1(y)∩Γ1(z)|=αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)| for all y∈Γ2(x) and z∈Γi(x)∩Γi(y). Furthermore, we prove Δ2=0 and either D=5 or c2∈{1,2}. We show there exist integers 3≤f≤ℓ≤D−2 such that dim(Ei∗W)=2 if and only if f≤i≤ℓ.

On bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint two

Let Γ denote a bipartite distance-regular graph with diameter D≥4 and valency k≥3. Let X denote the vertex set of Γ, and let A denote the adjacency matrix of Γ. For x∈X let T=T(x) denote the subalgebra of MatX(C) generated by A, E0⁎,E1⁎,…,ED⁎, where for 0≤i≤D, Ei⁎ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. An irreducible T-module W is said to be thin whenever dim Ei⁎W≤1 for 0≤i≤D. By the endpoint of W we mean min{i|Ei⁎W≠0}. For 0≤i≤D, let Γi(z) denote the set of vertices in X that are distance i from vertex z. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. In this paper we prove the following are equivalent: (i) Δ2>0 and for 2≤i≤D−2 there exist complex scalars αi,βi with the following property: for all x,y,z∈X such that ∂(x,y)=2, ∂(x,z)=i, ∂(y,z)=i we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|; (ii) For all x∈X there exist up to isomorphism exactly two irreducible modules for the Terwilliger algebra T(x) with endpoint two, and these modules are thin.

On the Terwilliger algebra of bipartite distance-regular graphs with [formula omitted] and [formula omitted

Let Γ denote a bipartite distance-regular graph with diameter D≥4 and valency k≥3. Let X denote the vertex set of Γ, and let A denote the adjacency matrix of Γ. For x∈X and for 0≤i≤D, let Γi(x) denote the set of vertices in X that are distance i from vertex x. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. It is known that Δ2=0 implies that D≤5 or c2∈{1,2}.For x∈X let T=T(x) denote the subalgebra of MatX(C) generated by A,E0∗,E1∗,…,ED∗, where for 0≤i≤D, Ei∗ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. By the endpoint of an irreducible T-module W we mean min{i|Ei∗W≠0}.We find the structure of irreducible T-modules of endpoint 2 for graphs Γ which have the property that for 2≤i≤D−1, there exist complex scalars αi, βi such that for all x,y,z∈X with ∂(x,y)=2,∂(x,z)=i,∂(y,z)=i, we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|, in case when Δ2=0 and c2=2. The case when Δ2=0 and c2=1 is already studied by MacLean et al. [15].We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2 and it is not thin. We give a basis for this T-module, and we give the action of A on this basis.

The attenuated space poset [formula omitted

In this paper, we study the incidence algebra T of the attenuated space poset Aq(N,M). We consider the following topics. We consider some generators of T: the raising matrix R, the lowering matrix L, and a certain diagonal matrix K. We describe some relations among R,L,K. We put these relations in an attractive form using a certain matrix S in T. We characterize the center Z(T). Using Z(T), we relate T to the quantum group Uτ(sl2) with τ2=q. We consider two elements A,A⁎ in T of a certain form. We find necessary and sufficient conditions for A,A⁎ to satisfy the tridiagonal relations. Let W denote an irreducible T-module. We find necessary and sufficient conditions for the above A,A⁎ to act on W as a Leonard pair.

On the Terwilliger algebra of bipartite distance-regular graphs with Δ2 = 0 and c2 = 1

Let Γ denote a bipartite distance-regular graph with diameter D≥4 and valency k≥3. Let X denote the vertex set of Γ, and let A denote the adjacency matrix of Γ. For x∈X and for 0≤i≤D, let Γi(x) denote the set of vertices in X that are distance i from vertex x. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. We first show that Δ2=0 implies that D≤5 or c2∈{1,2}.For x∈X let T=T(x) denote the subalgebra of MatX(C) generated by A,E0⁎,E1⁎,…,ED⁎, where for 0≤i≤D, Ei⁎ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. By the endpoint of an irreducible T-module W we mean min{i|Ei⁎W≠0}.In this paper we assume Γ has the property that for 2≤i≤D−1, there exist complex scalars αi, βi such that for all x,y,z∈X with ∂(x,y)=2, ∂(x,z)=i, ∂(y,z)=i, we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|. We additionally assume that Δ2=0 with c2=1.Under the above assumptions we study the algebra T. We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2. We give an orthogonal basis for this T-module, and we give the action of A on this basis.

Distance-regular graphs of q-Racah type and the universal Askey–Wilson algebra

Let C denote the field of complex numbers, and fix a nonzero q∈C such that q4≠1. Define a C-algebra Δq by generators and relations in the following way. The generators are A, B, C. The relations assert that each ofA+qBC−q−1CBq2−q−2,B+qCA−q−1ACq2−q−2,C+qAB−q−1BAq2−q−2 is central in Δq. The algebra Δq is called the universal Askey–Wilson algebra. Let Γ denote a distance-regular graph that has q-Racah type. Fix a vertex x of Γ and let T=T(x) denote the corresponding subconstituent algebra. In this paper we discuss a relationship between Δq and T. Assuming that every irreducible T-module is thin, we display a surjective C-algebra homomorphism Δq→T. This gives a Δq action on the standard module of T.

Dual polar graphs, the quantum algebra [formula omitted], and Leonard systems of dual q-Krawtchouk type

In this paper we consider how the following three objects are related: (i) the dual polar graphs; (ii) the quantum algebra Uq(sl2); (iii) the Leonard systems of dual q-Krawtchouk type. For convenience we first describe how (ii) and (iii) are related. For a given Leonard system of dual q-Krawtchouk type, we obtain two Uq(sl2)-module structures on its underlying vector space. We now describe how (i) and (iii) are related. Let Γ denote a dual polar graph. Fix a vertex x of Γ and let T=T(x) denote the corresponding subconstituent algebra. By definition T is generated by the adjacency matrix A of Γ and a certain diagonal matrix A*=A*(x) called the dual adjacency matrix that corresponds to x. By construction the algebra T is semisimple. We show that for each irreducible T-module W the restrictions of A and A* to W induce a Leonard system of dual q-Krawtchouk type. We now describe how (i) and (ii) are related. We obtain two Uq(sl2)-module structures on the standard module of Γ. We describe how these two Uq(sl2)-module structures are related. Each of these Uq(sl2)-module structures induces a C-algebra homomorphism Uq(sl2)→T. We show that in each case T is generated by the image together with the center of T. Using the combinatorics of Γ we obtain a generating set L,F,R,K of T along with some attractive relations satisfied by these generators.

Structure of thin irreducible modules of a Q-polynomial distance-regular graph

Let Γ be a Q-polynomial distance-regular graph with vertex set X, diameter D ⩾ 3 and adjacency matrix A. Fix x ∈ X and let A ∗ = A ∗ ( x ) be the corresponding dual adjacency matrix. Recall that the Terwilliger algebra T = T ( x ) is the subalgebra of Mat X ( C ) generated by A and A ∗ . Let W denote a thin irreducible T-module. It is known that the action of A and A ∗ on W induces a linear algebraic object known as a Leonard pair. Over the past decade, many results have been obtained concerning Leonard pairs. In this paper, we apply these results to obtain a detailed description of W. In our description, we do not assume that the reader is familiar with Leonard pairs. Everything will be proved from the point of view of Γ . Our results are summarized as follows. Let { E i } i = 0 D be a Q-polynomial ordering of the primitive idempotents of Γ and let { E i ∗ } i = 0 D be the dual primitive idempotents of Γ with respect to x. Let r , t and d be the endpoint, dual endpoint and diameter of W, respectively. Let u and v be nonzero vectors in E t W and E r ∗ W respectively. We show that { E r + i ∗ A i v } i = 0 d and { E t + i A * i u } i = 0 d are bases for W that are orthogonal with respect to the standard Hermitian dot product. We display the matrix representations of A and A ∗ with respect to these bases. We associate with W two sequences of polynomials { p i } i = 0 d and { p i ∗ } i = 0 d . We show that for 0 ⩽ i ⩽ d , p i ( A ) v = E r + i ∗ A i v and p i ∗ ( A ∗ ) u = E t + i A * i u . Next, we show that { E r + i ∗ u } i = 0 d and { E t + i v } i = 0 d are orthogonal bases for W ; we call these the standard basis and dual standard basis for W , respectively. We display the matrix representations of A and A ∗ with respect to these bases. The entries in these matrices will play an important role in our theory. We call these the intersection numbers and dual intersection numbers of W . Using these numbers, we compute all inner products involving the standard and dual standard bases. We also use these numbers to define two normalizations u i , v i (resp. u i ∗ , v i ∗ ) for p i (resp. p i ∗ ). Using the orthogonality of the standard and dual standard bases, we show that for each of the sequences { p i } i = 0 d , { p i ∗ } i = 0 d , { u i } i = 0 d , { u i ∗ } i = 0 d , { v i } i = 0 d , { v i ∗ } i = 0 d the polynomials involved are orthogonal and we display the orthogonality relations. We also show that each of the sequences satisfy a three-term recurrence and a relation known as the Askey–Wilson duality. We then turn our attention to two more bases for W . We find the matrix representations of A and A ∗ with respect to these bases. From the entries of these matrices we obtain two sequences of scalars known as the first split sequence and second split sequence of W . We associate with W a sequence of scalars called the parameter array. This sequence consists of the eigenvalues of the restriction of A to W, the eigenvalues of the restriction of A ∗ to W , the first split sequence of W and the second split sequence of W. We express all the scalars and polynomials associated with W in terms of its parameter array. We show that the parameter array of W is determined by r , t , d and one more free parameter. From this we conclude that the isomorphism class of W is determined by these four parameters. Finally, we apply our results to the case in which Γ has q-Racah type or classical parameters.

Leonard triples and hypercubes

Let V denote a vector space over ? with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear operators on V such that for each of these operators there exists a basis of V with respect to which the matrix representing that operator is diagonal and the matrices representing the other two operators are irreducible tridiagonal. Let D denote a positive integer and let Q D denote the graph of the D-dimensional hypercube. Let X denote the vertex set of Q D and let $A\in {\rm Mat}_{X}({\mathbb{C}})$ denote the adjacency matrix of Q D . Fix x?X and let $A^{*}\in {\rm Mat}_{X}({\mathbb{C}})$ denote the corresponding dual adjacency matrix. Let T denote the subalgebra of ${\rm Mat}_{X}({\mathbb{C}})$ generated by A,A *. We refer to T as the Terwilliger algebra of Q D with respect to x. The matrices A and A * are related by the fact that 2i A=A * A ? ?A ? A * and 2i A *=A ? A?AA ? , where 2i A ? =AA *?A * A and i 2=?1. We show that the triple A, A *, A ? acts on each irreducible T-module as a Leonard triple. We give a detailed description of these Leonard triples.

The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two

We consider a bipartite distance-regular graph Γ with diameter D ⩾ 4 , valency k ⩾ 3 , intersection numbers b i , c i , distance matrices A i , and eigenvalues θ 0 > θ 1 > ⋯ > θ D . Let X denote the vertex set of Γ and fix x ∈ X . Let T = T ( x ) denote the subalgebra of Mat X ( C ) generated by A , E 0 * , E 1 * , … , E D * , where A = A 1 and E i * denotes the projection onto the i th subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever dim E i * W ⩽ 1 for 0 ⩽ i ⩽ D . By the endpoint of W we mean min { i | E i * W ≠ 0 } . Assume W is thin with endpoint 2. Observe E 2 * W is a one-dimensional eigenspace for E 2 * A 2 E 2 * ; let η denote the corresponding eigenvalue. It is known θ ˜ 1 ⩽ η ⩽ θ ˜ d where θ ˜ 1 = - 1 - b 2 b 3 ( θ 1 2 - b 2 ) - 1 , θ ˜ d = - 1 - b 2 b 3 ( θ d 2 - b 2 ) - 1 , and d = ⌊ D / 2 ⌋ . To describe the structure of W we distinguish four cases: (i) η = θ ˜ 1 ; (ii) D is odd and η = θ ˜ d ; (iii) D is even and η = θ ˜ d ; (iv) θ ˜ 1 < η < θ ˜ d . We investigated cases (i), (ii) in MacLean and Terwilliger [Taut distance-regular graphs and the subconstituent algebra, Discrete Math. 306 (2006) 1694–1721]. Here we investigate cases (iii), (iv) and obtain the following results. We show the dimension of W is D - 1 - e where e = 1 in case (iii) and e = 0 in case (iv). Let v denote a nonzero vector in E 2 * W . We show W has a basis E i v ( i ∈ S ) , where E i denotes the primitive idempotent of A associated with θ i and where the set S is { 1 , 2 , … , d - 1 } ∪ { d + 1 , d + 2 , … , D - 1 } in case (iii) and { 1 , 2 , … , D - 1 } in case (iv). We show this basis is orthogonal (with respect to the Hermitian dot product) and we compute the square-norm of each basis vector. We show W has a basis E i + 2 * A i v ( 0 ⩽ i ⩽ D - 2 - e ) , and we find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square-norm of each basis vector. We find the transition matrix relating our two bases for W.

Taut distance-regular graphs and the subconstituent algebra

We consider a bipartite distance-regular graph Γ with diameter D ⩾ 4 and valency k ⩾ 3 . Let X denote the vertex set of Γ and fix x ∈ X . Let Γ 2 2 denote the graph with vertex set X ˘ = { y ∈ X | ∂ ( x , y ) = 2 } , and edge set R ˘ = { yz | y , z ∈ X ˘ , ∂ ( y , z ) = 2 } , where ∂ is the path-length distance function for Γ . The graph Γ 2 2 has exactly k 2 vertices, where k 2 is the second valency of Γ . Let η 1 , η 2 , … , η k 2 denote the eigenvalues of the adjacency matrix of Γ 2 2 ; we call these the local eigenvalues of Γ . Let A denote the adjacency matrix of Γ . We obtain upper and lower bounds for the local eigenvalues in terms of the intersection numbers of Γ and the eigenvalues of A. Let T = T ( x ) denote the subalgebra of Mat X ( C ) generated by A , E 0 * , E 1 * , … , E D * , where for 0 ⩽ i ⩽ D , E i * represents the projection onto the i th subconstituent of Γ with respect to x. We refer to T as the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever dim E i * W ⩽ 1 for 0 ⩽ i ⩽ D . By the endpoint of W we mean min { i | E i * W ≠ 0 } . We give a detailed description of the thin irreducible T-modules that have endpoint 2 and dimension D - 3 . MacLean [An inequality involving two eigenvalues of a bipartite distance-regular graph, Discrete Math. 225 (2000) 193–216] defined what it means for Γ to be taut. We obtain three characterizations of the taut condition, each of which involves the local eigenvalues or the above T-modules.

The Terwilliger Algebra of a Distance-Regular Graph that Supports a Spin Model

Let ? denote a distance-regular graph with vertex set X, diameter D ? 3, valency k ? 3, and assume ? supports a spin model W. Write W = ?i = 0D ti Ai where Ai is the ith distance-matrix of ?. To avoid degenerate situations we assume ? is not a Hamming graph and ti ? {t0, ?t0 } for 1 ? i ? D. In an earlier paper Curtin and Nomura determined the intersection numbers of ? in terms of D and two complex parameters ? and q. We extend their results as follows. Fix any vertex x ? X and let T = T(x) denote the corresponding Terwilliger algebra. Let U denote an irreducible T-module with endpoint r and diameter d. We obtain the intersection numbers ci(U), bi(U), ai(U) as rational expressions involving r, d, D, ? and q. We show that the isomorphism class of U as a T-module is determined by r and d. We present a recurrence that gives the multiplicities with which the irreducible T-modules appear in the standard module. We compute these multiplicites explicitly for the irreducible T-modules with endpoint at most 3. We prove that the parameter q is real and we show that if ? is not bipartite, then q > 0 and ? is real.

An Inequality Involving the Local Eigenvalues of a Distance-Regular Graph

Let Γ denote a distance-regular graph with diameter D ≥ 3, valency k, and intersection numbers a i, b i, c i. Let X denote the vertex set of Γ and fix x ∈ X. Let Δ denote the vertex-subgraph of Γ induced on the set of vertices in X adjacent X. Observe Δ has k vertices and is regular with valency a 1. Let η1 ≥ η2 ≥ ··· ≥ η k denote the eigenvalues of Δ and observe η1 = a 1. Let Φ denote the set of distinct scalars among η2, η3, ..., η k . For η ∈ Φ let multη denote the number of times η appears among η2, η3,...,η k . Let λ denote an indeterminate, and let p 0, p1, ...,p D denote the polynomials in \(\mathbb{R}\)[λ] satisfying p 0 = 1 andλp i = c i+1 p i+1 + (a i − c i+1 + c i)p i + b i p i−1 (0 ≤ i ≤ D − 1),where p −1 = 0. We show EquationSource % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0de9qqFf0x% c9q8qqaqFn0dXdir-xcvk9ar-Jbba9q8Gq0-yq-He9q8qqQ8frFve9% Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIXaGaey% 4kaSYaaabuaeaadaWcaaqaaiaadcfadaWgaaqcbaAaaiaadMgacqGH% sislcaaIXaaaleqaaOGaaiikaiqbeE7aOzaaiaGaaiykaaqaaiaadc% fadaWgaaqcbaAaaiaadMgaaSqabaGccaGGOaGafq4TdGMbaGaacaGG% PaGaaiikaiaaigdacqGHRaWkcuaH3oaAgaacaiaacMcaaaaalqaabe% qaaiabeE7aOjabgIGiolabfA6agbqaaiabeE7aOjabgcMi5kabgkHi% Tiaaigdaaaqab0GaeyyeIuoakiaabccacaqGTbGaaeyDaiaabYgaca% qG0bWaaSbaaSqaaiabeE7aObqabaGccqGHKjYOdaWcaaqaaiaadUga% aeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaaaakiaabccacaqGOaGaae% ymaiabgsMiJkaadMgacqGHKjYOcaWGebGaeyOeI0IaaGymaiaabMca% caqGSaaaaa!6BFB! where we abbreviate \(\tilde \eta \) = −1 − b 1(1+η)−1. Concerning the case of equality we obtain the following result. Let T = T(x) denote the subalgebra of Mat X (\(\mathbb{C}\)) generated by A, E*0, E*1, ..., E* D , where A denotes the adjacency matrix of Γ and E* i denotes the projection onto the ith subconstituent of Γ with respect to X. T is called the subconstituent algebra or the Terwilliger algebra. An irreducible T-module W is said to be thin whenever dimE* i W ≤ 1 for 0 ≤ i ≤ D. By the endpoint of W we mean min{i|E* i W ≠ 0}. We show the following are equivalent: (i) Equality holds in the above inequality for 1 ≤ i ≤ D − 1; (ii) Equality holds in the above inequality for i = D − 1; (iii) Every irreducible T-module with endpoint 1 is thin.

Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra

Let Γ denote a distance-regular graph with diameter D≥3, intersection numbers a i , b i , c i and Bose–Mesner algebra M. For θ∈ C∪∞ we define a one-dimensional subspace of M which we call M( θ). If θ∈ C then M( θ) consists of those Y in M such that (A−θI)Y∈ CA D , where A (resp. A D ) is the adjacency matrix (resp. Dth distance matrix) of Γ. If θ=∞ then M (θ)= CA D . By a pseudo primitive idempotent for θ we mean a nonzero element of M( θ). We use these as follows. Let X denote the vertex set of Γ and fix x∈ X. Let T denote the subalgebra of Mat X( C) generated by A, E 0 ∗,E 1 ∗,…,E D ∗ , where E i ∗ denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the Terwilliger algebra. Let W denote an irreducible T-module. By the endpoint of W we mean min{i∣E i ∗W≠0} . W is called thin whenever dim(E i ∗W)≤1 for 0≤ i≤ D. Let V= C X denote the standard T-module. Fix 0≠v∈E 1 ∗V with v orthogonal to the all ones vector. We define ( M ;v):={P∈ M ∣Pv∈E D ∗V} . We show the following are equivalent: (i) dim( M; v)≥2; (ii) v is contained in a thin irreducible T-module with endpoint 1. Suppose (i), (ii) hold. We show ( M; v) has a basis J, E where J has all entries 1 and E is defined as follows. Let W denote the T-module which satisfies (ii). Observe E 1 ∗W is an eigenspace for E 1 ∗AE 1 ∗ ; let η denote the corresponding eigenvalue. Define η=−1−b 1(1+η) −1 if η≠−1 and η=∞ if η=−1. Then E is a pseudo primitive idempotent for η .

The subconstituent algebra of a distance-regular graph; thin modules with endpoint one

We consider a distance-regular graph Γ with diameter D⩾3, intersection numbers a i , b i , c i and eigenvalues θ 0> θ 1>⋯> θ D . Let X denote the vertex set of Γ and fix x∈ X. Let T= T( x) denote the subalgebra of Mat X( C) generated by A, E * 0, E * 1,…, E * D , where A denotes the adjacency matrix of Γ and E * i denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever dim E * iW⩽1 for 0⩽ i⩽ D. By the endpoint of W we mean min{i | E * iW≠0} . We describe the thin irreducible T-modules with endpoint 1. Let W denote a thin irreducible T-module with endpoint 1. Observe E * 1 W is a one-dimensional eigenspace for E * 1 AE * 1; let η denote the corresponding eigenvalue. It is known θ ̃ 1⩽η⩽ θ ̃ D where θ ̃ 1=−1−b 1(1+θ 1) −1 and θ ̃ D=−1−b 1(1+θ D) −1 . For η= θ ̃ 1 and η= θ ̃ D the structure of W was worked out by Go and the present author [Tight distance-regular graphs and the subconstituent algebra, preprint]. For θ ̃ 1<η< θ ̃ D we obtain the following results. We show the dimension of W is D. Let v denote a nonzero vector in E * 1 W. We show W has a basis E i v (1⩽ i⩽ D), where E i denotes the primitive idempotent of A associated with θ i . We show this basis is orthogonal (with respect to the Hermitian dot product) and we compute the square norm of each basis vector. We show W has a basis E * i+1 A i v (0⩽ i⩽ D−1), where A i denotes the ith distance matrix for Γ. We find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square norm of each basis vector. We find the transition matrix relating our two bases for W.