## The subconstituent algebra of a strongly regular graph

##### J. Algebr. Comb. 22(1), 5-38 (2005)
DOI: 10.1007/s10801-005-2504-4

### Summary

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/Phrases

keywords distance-regular graph, association scheme, subconstituent algebra, Terwilliger algebra, tight graph, completely regular code