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.

令\(\Gamma \)是一棵有限树。固定碱顶点\（X_0 \）的\（\伽玛\） ，并让\（T = T ^ {（X_0）} \）是的特威利格代数\（\伽玛\）相对于\（X_0 \） . 用H表示固定\(x_0\)的\(\Gamma \)自同构群，让\(S={\mathrm{End}}_H~(V)\)是H的中心化代数，其中\(V={\mathbb {C}}X\)是T的标准模块，其中X是\(\Gamma \)的底层顶点集。很明显，T包含在S 中。我们通过比较它们的不可约表示来展示T和S之间的差距有多大；特别是当\(T=S\)成立时，我们可以精确地找到。