Irreducible Representations of the Terwilliger Algebra of a Tree

Let Γ be a finite tree. Fix a base vertex x_0 of Γ and let T=T^(x_0) be the Terwilliger algebra of Γ with respect to x_0 . Denote by H the group of automorphisms of Γ that fix x_0 , and let S=End_H (V) be the centralizer algebra of H , where V=ℂX is the standard module of T with X the underlying vertex set of Γ . 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.