Tight distance-regular graphs and the subconstituent algebra

We consider a distance-regular graph Gamma with diameter D greater than or equal to 3, intersection numbers a(i), b(i), c(i) and eigenvalues k = theta(0) > theta(1) >...> theta(D). Let X denote the vertex set of Gamma and fix x is an element of 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 Gamma and E-i(*) denotes the projection onto the ith subconstituent of P with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Gamma with respect to x. An irreducible T-module W is said to be thin whenever dim E-i(*) W less than or equal to 1 for 0 less than or equal to i less than or equal to D. By the endpoint of W we mean min{i\E-i(*) W not equal 0}. 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 eta denote the corresponding eigenvalue. We call eta the local eigenvalue of W. It is known (theta) over tilde (1) less than or equal to eta less than or equal to (theta) over tilde (D) where (theta) over tilde (1) = -1 - b(1) (1 + theta(1))(-1) and (theta) over tilde (D) = -1 - b(1) (1 + theta(D))(-1). Let n = 1 or n = D and assume eta = (theta) over tilde (n). We show the dimension of W is D-1. Let upsilon denote a nonzero vector in E-1(*) W. We show W has a basis E(i)upsilon (1 less than or equal to i less than or equal to D, i :A n), where E-i denotes the primitive idempotent of A associated with theta(i). We show this basis is orthogonal (with respect to the Hermitean dot product) and we compute the square norm of each basis vector. We show W has a basis E-i+1(*), A(i)upsilon (0 less than or equal to i less than or equal to D - 2), where A(i) denotes the ith distance matrix for Gamma. 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. For notational convenience, we say Gamma is 1-thin with respect to x whenever every irreducible T-module with endpoint 1 is thin. Similarly, we say Gamma is tight with respect to x whenever every irreducible T-module with endpoint 1 is thin with local eigenvalue (theta) over tilde (1) or (theta) over tilde (D). In [J Algebr Comb., 12, (2000), 163-197] Jurisic, Koolen and Terwilliger showed (theta(1) + k/a(1)+1)(theta(D) + k/a(1)+1) greater than or equal to -ka(1)b(1)/(a(1)+1)(2). They defined Gamma to be tight whenever Gamma is nonbipartite and equality holds above. We show the following are equivalent: (i) Gamma is tight; (ii) Gamma is tight with respect to each vertex; (iii) Gamma is tight with respect to at least one vertex. We show the following are equivalent: (i) Gamma is tight; (ii) Gamma is nonbipartite, a(D) = 0, and Gamma is 1-thin with respect to each vertex; (iii) Gamma is nonbipartite, a(D) = 0, and Gamma is 1-thin with respect to at least one vertex. (C) 2002 Elsevier Science Ltd. All rights reserved.