The Terwilliger algebras of Johnson graphs

Abstract

Let $J(n,m)$ denote the Johnson graph with vertex set X. Fix a vertex $x\in X$. Let $T=T(x)$ denote the Terwilliger algebra of $J(n,m)$ corresponding to x. In this paper we study the structure of T under the assumption that $m⩾3$ and $n⩾4m$. Let $U({\mathit{sl}}_2)$ be the universal enveloping algebra of ${\mathit{sl}}_2$. We display a $\mathbb{C}$-algebra homomorphism $\vartheta :U({\mathit{sl}}_2)\to T$. We show that T is generated by the image of ϑ and some central elements of T. We obtain these results using the theory of Leonard pairs.