The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph

Let Γ be a distance-regular graph of diameter D. Let X denote the vertex set of Γ and let Y be a nonempty subset of X. We define an algebra τ = τ(Y). This algebra is finite dimensional and semisimple. If Y consists of a single vertex then τ is the corresponding subconstituent algebra defined by P. Terwilliger. We investigate the irreducible τ-modules. We define endpoints and thin condition on irreducible τ-modules as a generalization of the case when Y consists of a single vertex. We determine when an irreducible module is thin. When the module is generated by the characteristic vector of Y, it is thin if and only if Y is a completely regular code of Γ. By considering a suitable subset Y, every irreducible τ(x)-module of endpoint i can be regarded as an irreducible τ(Y)-module of endpoint 0.This research was partially supported by the Grant-in-Aid for Scientific Research (No. 12640039), Japan Society of the Promotion of Science. A part of the research was done when the author was visiting the Ohio State University.     

Twice Q-polynomial distance-regular graphs are thin

Let Γ be a distance-regular graph of diameter d 3. For each vertex χ of Γ, let T(χ) denote the subconstituent algebra for Γ with respect to χ. An irreducible T(χ)-module W is said to be thin if dim E_i^*(χ) W 1 for 0 i d, where E_i^*(χ) is the projection onto the ith subconstituent for Γ with respect to χ. The graph Γ is said to be thin if, for each vertex χ of Γ, very irreducible T(χ)-module is thin. Our main result is the following Theorem: If Γ has two Q-polynomial structures, then Γ is thin.     

The girth of a thin distance-regular graph

Let Γ be a distance-regular graph of diameterd≥3. For each vertexx of Γ, letT(x) denote the Terwilliger algebra for Γ with respect tox. An irreducibleT(x)-moduleW is said to bethin if dimE _i ^* (x)W≤1 for 0≤i≤d, whereE _i ^* (x) is theith dual idempotent for Γ with respect tox. The graph Γ isthin if for each vertexx of Γ, every irreducibleT(x)-module is thin. Aregular generalized quadrangle is a bipartite distance-regular graph with girth 8 and diameter 4. Our main results are as follows: Theorem. Let Γ=(X,R) be a distance-regular graph with diameter d≥3 and valency k≥3. Then the following are equivalent:

1. Γis a regular generalized quadrangle.
2. Γis thin and c ₃=1.

Corollary. Let Γ=(X,R) be a thin distance-regular graph with diameter d≥3 and valency k≥3. Then Γ has girth 3, 4, 6, or 8. Then girth of Γ is 8 exactly when Γ is a regular generalized quadrangle.     

The Subconstituent Algebra of an Association Scheme, (Part I)

We introduce a method for studying commutative association schemes with many vanishing intersection numbers and/or Krein parameters, and apply the method to the P- and Q-polynomial schemes. Let Y denote any commutative association scheme, and fix any vertex x of Y. We introduce a non-commutative, associative, semi-simple -algebra T = T(x) whose structure reflects the combinatorial structure of Y. We call T the subconstituent algebra of Y with respect to x. Roughly speaking, T is a combinatorial analog of the centralizer algebra of the stabilizer of x in the automorphism group of Y.In general, the structure of T is not determined by the intersection numbers of Y, but these parameters do give some information. Indeed, we find a relation among the generators of T for each vanishing intersection number or Krein parameter.We identify a class of irreducible T-moduIes whose structure is especially simple, and say the members of this class are thin. Expanding on this, we say Y is thin if every irreducible T(y)-module is thin for every vertex y of Y. We compute the possible thin, irreducible T-modules when Y is P- and Q-polynomial. The ones with sufficiently large dimension are indexed by four bounded integer parameters. If Y is assumed to be thin, then sufficiently large dimension means dimension at least four.We give a combinatorial characterization of the thin P- and Q-polynomial schemes, and supply a number of examples of these objects. For each example, we show which irreducible T-modules actually occur.We close with some conjectures and open problems.     

Bipartite Distance-regular Graphs, Part I

Let Γ=(X,E) denote a bipartite distance-regular graph with diameter D≥4, and fix a vertex x of Γ. The Terwilliger algebra T=T(x) is the subalgebra of Mat _X(C) generated by A, E ^* ₀, E ^* ₁,…,E ^* _D, where A denotes the adjacency matrix for Γ and E ^* _i denotes the projection onto the i ^TH subconstituent of Γ with respect to x. An irreducible T-module W is said to be thin whenever dimE ^* _i W≤1 for 0≤i≤Di. The endpoint of W is min{i|E ^* _i W≠0}. We determine the structure of the (unique) irreducible T-module of endpoint 0 in terms of the intersection numbers of Γ. We show that up to isomorphism there is a unique irreducible T-module of endpoint 1 and it is thin. We determine its structure in terms of the intersection numbers of Γ. We determine the structure of each thin irreducible T-module W of endpoint 2 in terms of the intersection numbers of Γ and an additional real parameter ψ=ψ(W), which we refer to as the type of W. We now assume each irreducible T-module of endpoint 2 is thin and obtain the following two-fold result. First, we show that the intersection numbers of Γ are determined by the diameter D of Γ and the set of ordered pairs

The Terwilliger algebra of the incidence graph of the Hamming graph

Let \(\varGamma \) be a distance-semiregular graph on Y, and let \(D^Y\) be the diameter of \(\varGamma \) on Y. Let \(\varDelta \) be the halved graph of \(\varGamma \) on Y. Fix \(x \in Y\). Let T and \(T'\) be the Terwilliger algebras of \(\varGamma \) and \(\varDelta \) with respect to x, respectively. Assume, for an integer i with \(1 \le 2i \le D^Y\) and for \(y,z \in \varGamma _{2i}(x)\) with \(\partial _{\varGamma }(y,z)=2\), the numbers \(|\varGamma _{2i-1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) and \(|\varGamma _{2i+1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) depend only on i and do not depend on the choice of y, z. The first goal in this paper is to show the relations between T-modules of \(\varGamma \) and \(T'\)-modules of \(\varDelta \). Assume \(\varGamma \) is the incidence graph of the Hamming graph H(D, n) on the vertex set Y and the set \({\mathcal {C}}\) of all maximal cliques. Then, \(\varGamma \) satisfies above assumption and \(\varDelta \) is isomorphic to H(D, n). The second goal is to determine the irreducible T-modules of \(\varGamma \). For each irreducible T-module W, we give a basis for W the action of the adjacency matrix on this basis and we calculate the multiplicity of W.     

Distance-Regular Graphs Related to the Quantum Enveloping Algebra of sl(2)

We investigate a connection between distance-regular graphs and U _q(sl(2)), the quantum universal enveloping algebra of the Lie algebra sl(2). Let be a distance-regular graph with diameter d 3 and valency k 3, and assume is not isomorphic to the d-cube. Fix a vertex x of , and let (x) denote the Terwilliger algebra of with respect to x. Fix any complex number q {0, 1, –1}. Then is generated by certain matrices satisfying the defining relations of U _q(sl(2)) if and only if is bipartite and 2-homogeneous.     

Bipartite Distance-regular Graphs, Part II

This is a continuation of “Bipartite Distance-regular Graphs, Part I”. We continue our study of the Terwilliger algebra T of a bipartite distance-regular graph. In this part we focus on the thin irreducible T-modules of endpoint 2 and on those distance-regular graphs for which every irreducible T-module of endpoint 2 is thin. Revised: June 2, 1997     

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 = 0}^D t_i A_i where A_i is the ith distance-matrix of . To avoid degenerate situations we assume is not a Hamming graph and t_i {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 c_i(U), b_i(U), a_i(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.AMS 2000 Subject Classification: Primary 05E30     

An Application of U(g)-bimodules to Representation Theory of Affine Lie Algebras

Let be the affine Lie algebra associated to the simple finite-dimensional Lie algebra . We consider the tensor product of the loop -module associated to the irreducible finite-dimensional -module V() and the irreducible highest weight -module L _k,. Then L _k, can be viewed as an irreducible module for the vertex operator algebra M _k,0. Let A(L _k,) be the corresponding -bimodule. We prove that if the -module is zero, then the -module is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.     

Vertex operator algebras,generalized doubles and dual pairs

Let V be a simple vertex operator algebra and G a finite automorphism group. Then there is a natural right G-action on the set of all inequivalent irreducible V-modules. Let be a finite set of inequivalent irreducible V-modules which is closed under the action of G. There is a finite dimensional semisimple associative algebra for a suitable 2-cocycle naturally determined by the G-action on such that and the vertex operator algebra form a dual pair on the sum of V-modules in in the sense of Howe. In particular, every irreducible V-module is completely reducible -module. Received: 10 September, 2001 / Published online: 29 April 2002 RID="*" ID="*" Supported by NSF grants and a research grant from the Committee on Research, UC Santa Cruz. RID="**" ID="**" Supported by DPST grant from government of Thailand.     

The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4

In a cyclotomic scheme over a finite field, there are some relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the field. These relations were investigated in [3]. In this paper, we replace the finite field by a commutative local ring which is called a Galois ring of characteristic 4. Hence we want to find similar relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the local ring. Specifically, if we let be a Galois ring of characteristic 4,X a cyclotomic scheme over with classD and the Terwilliger algebra ofX, then we show that most of the irreducible -modules have standard forms; otherwise, certain relations of the Jacobi sums hold. When the classD is three, we can completely determine the irreducible -modules using Jacobi sums.     

A proof of Freud's conjecture for exponential weights

LetW (x) be a function nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ²(x) are finite. Let {p _n (W ²;x)} ₀ denote the sequence of orthonormal polynomials with respect to the weightW ²(x), and let {A _n } ₁ and {B _n } ₁ denote the coefficients in the recurrence relation

Markov Processes with Equal Capacities

Let X and be transient standard Markov processes in weak duality with respect to a -finite measure m. Let (Y, , ) be a second dual pair with the same state space E as (X, , m). Let Cap ^X and Cap ^Y be the 0-order capacities associated with (X, , m) and (Y, , ), and let V and denote the potential kernels for Y and . Assume that singletons are polar with respect to both X and Y, and that semipolar sets are of capacity zero for both dual pairs. We show that if Cap ^X (B)=Cap ^Y (B) for every Borel subset of E then there is a strictly increasing continuous additive functional D=(D _t) _t0 of (X, , m) such that

Completely regular clique graphs,II

Let \(\varGamma = (X,R)\) be a connected graph. Then \(\varGamma \) is said to be a completely regular clique graph of parameters (s, c) with \(s\ge 1\) and \(c\ge 1\), if there is a collection \({\mathcal {C}}\) of completely regular cliques of size \(s+1\) such that every edge is contained in exactly c members of \({\mathcal {C}}\). In the previous paper (Suzuki in J Algebr Combin 40:233–244, 2014), we showed, among other things, that a completely regular clique graph is distance-regular if and only if it is a bipartite half of a certain distance-semiregular graph. In this paper, we show that a completely regular clique graph with respect to \({\mathcal {C}}\) is distance-regular if and only if every \({\mathcal {T}}(C)\)-module of endpoint zero is thin for all \(C\in {\mathcal {C}}\). We also discuss the relation between a \({\mathcal {T}}(C)\)-module of endpoint 0 and a \({\mathcal {T}}(x)\)-module of endpoint 1 and study examples of completely regular clique graphs.     

Tree coloring of distance graphs with a real interval set

Let R be the set of real numbers and D be a subset of the positive real numbers. The distance graph G(R,D) is a graph with the vertex set R and two vertices x and y are adjacent if and only if |x−y|D. In this work, the vertex arboricity (i.e., the minimum number of subsets into which the vertex set V(G) can be partitioned so that each subset induces an acyclic subgraph) of G(R,D) is determined for D being an interval between 1 and δ.     

On p-chief factors and extensions of $ \mathbb{K} $ G-modules

The subject of this paper is the relationship between the set of chief factors of a finite group G and extensions of an irreducible \mathbbK \mathbb{K} G-module U ( \mathbbK \mathbb{K} a field). Let H / L be a p-chief factor of G. We prove that, if H / L is complemented in a vertex of U, then there is a short exact sequence of Ext-functors for the module U and any \mathbbK \mathbb{K} G-module V. In some special cases, we prove the converse, which is false in general. We also consider the intersection of the centralizers of all the extensions of U by an irreducible module and provide new bounds for this group.     

On the distribution of integer points of rational curves

Let F(X,Y) be an absolutely irreducible polynomial in such that the algebraic curve C: F(X,Y) = 0 has infinitely many integer points. In this paper we obtain an explicit estimate on the distribution of integer points of C.     

Strongly consistent estimators of k-th order regression curves and rates of convergence

Summary Let X and Y be two jointly distributed real valued random variables, and let the conditional distribution of X given Y be either in a Lebesgue exponential family or in a discrete exponential family. Let r_k be the k-th order regression curve of Y on X. Let X _n=(X ₁,..., X_n) be a random sample of size n on X. For a subset S of the real line R, statistics based on X_n are exhibited and sufficient conditions are given under which is close to O(n ^–1/2) with probability one. To obtain this result, with uf (u known and f unknown) denoting the unconditional (on y) density of X, the problem of estimating r _k (·) is reduced to the one of estimating f ^(k) (·)/f(·) if the density is wrt the Lebesgue measure on R and f ^(k) is the k-th order derivative of f; and to the one of estimating f(·+k)/f(·) if the density is wrt the counting measure on a countable subset of R.