The subconstituent algebra of a strongly regular graph
J. Algebr. Comb. 22(1), 5-38 (2005)
Summary: Let $\Gamma $be a distance-regular graph of diameter $D$. Let $X$ denote the vertex set of $\Gamma $and let $Y$ be a nonempty subset of $X$. We define an algebra $\tau = \tau ( Y)$. This algebra is finite dimensional and semisimple. If $Y$ consists of a single vertex then $\tau $is the corresponding subconstituent algebra defined by P. Terwilliger. We investigate the irreducible $\tau $-modules. We define endpoints and thin condition on irreducible $\tau $-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 $\Gamma $. By considering a suitable subset $Y$, every irreducible $\tau ( x)$-module of endpoint $i$ can be regarded as an irreducible $\tau ( Y)$-module of endpoint 0.
keywords distance-regular graph, association scheme, subconstituent algebra, Terwilliger algebra, tight graph, completely regular code