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

Let Gamma denote a distance-regular graph with diameter D greater than or equal to 3, intersection numbers a(i), b(i), c(i) and Bose-Mesner algebra M. For theta is an element of C boolean OR infinity we define a one-dimensional subspace of M which we call M(theta). If theta is an element of C then M(theta) consists of those Y in M such that (A - thetaI)Y is an element of CA(D), where A (resp. A(D)) is the adjacency matrix (resp. Dth distance matrix) of Gamma. If theta = infinity then M(theta) = CA(D). By a pseudo primitive idempotent for theta we mean a nonzero element of M(theta). We use these as follows. Let X denote the vertex set of Gamma and fix x is an element of 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 Gamma with respect to x. T is called the Terwilliger algebra. Let W denote an irreducible T-module. By the end point of W we mean min{i \ (EiW)-W-* not equal 0}. W is called 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. Let V = C-X denote the standard T-module. Fix 0 not equal v is an element of (E1V)-V-* with v orthogonal to the all ones vector. We define (M; v) := {P is an element of M \ Pv is an element of (EDV)-V-*}. We show the following are equivalent: (i) dim(M; v) greater than or equal to 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 I and E is defined as follows. Let W denote the T-module which satisfies (ii). Observe (E1W)-W-* is an eigenspace for E(1)(*)AE(1)(*); let 17 denote the corresponding eigenvalue. Define = (η) over tilde = -1 - b(1) (1 + eta)(-1) if eta not equal -1 and infinity if eta = -1. Then E is a pseudo primitive idempotent for. (C) 2003 Elsevier Ltd. All rights reserved.