Irreducible Representations of the Terwilliger Algebra of a Tree

Abstract

Let $\Gamma$ be a finite tree. Fix a base vertex $x_0$ of $\Gamma$ and let $T=T^{(x_0)}$ be the Terwilliger algebra of $\Gamma$ with respect to $x_0$. Denote by H the group of automorphisms of $\Gamma$ that fix $x_0$, and let $S={\mathrm{End}}_H~(V)$ be the centralizer algebra of H, where $V={\mathbb {C}}X$ is the standard module of T with X the underlying vertex set of $\Gamma$. It is obvious that T is contained in S. We show how large the gap is between T and S by comparing irreducible representations of them; in particular we find precisely when $T=S$ holds.