The Terwilliger algebra for the distance-regular graphs with valency three
In this paper, we discuss a family of highly regular graphs, said to be distance-regular. We are particularly interested in the distance-regular graphs with valency three. It is known that there exist exactly 13 such graphs. Let $\Gamma$ denote a distance-regular graph with vertex set $X$. For any vertex $x \in X$, the corresponding Terwilliger algebra $T=T(x)$ is generated by the adjacency algebra $M$ of $\Gamma$ and the dual adjacency algebra $M^*=M^*(x)$ of $\Gamma$ with respect to $x$. It is known that the algebra $T$ is semisimple. By construction, the vector space $V=\mathbb{C}^X$ is a module for $T$, said to be standard. In this paper we have the following goal. For each of the 13 distance-regular graphs $\Gamma$ with valency three, we will decompose the standard module $V$ into a direct sum of irreducible $T$-modules. Using this information, we will work out the dimension of $T$.