The Terwilliger algebra of a distance-regular graph of negative type

Abstract

Let Γ denote a distance-regular graph with diameter D ≥ 3. Assume Γ has classical parameters (D, b, α, β) with b < −1. Let X denote the vertex set of Γ and let A ∈ Mat X (C) denote the adjacency matrix of Γ. Fix x ∈ X and let A * ∈ Mat X (C) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of Mat X (C) generated by A, A * . We call T the Terwilliger algebra of Γ with respect to x. We show that up to isomorphism there exist exactly two irreducible T -modules with endpoint 1; their dimensions are D and 2D − 2. For these T -modules we display a basis consisting of eigenvectors for A * , and for each basis we give the action of A. * Supported in part by "Javna agencija za raziskovalno dejavnost Republike Slovenije", program no. Z1-9614.

On the shape of a tridiagonal pair

“…We refer the reader to [1-3, 28, 37-40, 45-49, 51, 63-65, 73-78, 96, 98] for background information about tridiagonal pairs. See [29][30][31][32][33][34][35][36][41][42][43][44]50,[52][53][54][55][56][57][58][59][60][61][62][66][67][68][69][70][71][72][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93][94][95][96][97]…”