The Terwilliger algebras of Grassmann graphs

Let Jq(n,m)Jq(n,m) denote the Grassmann graph with vertex set X   and diameter min⁡{m,n−m}min⁡{m,n−m}. Fix a vertex x∈Xx∈X. Let T=T(x)T=T(x) denote the Terwilliger algebra of Jq(n,m)Jq(n,m) corresponding to x. In this paper we study the structure of T   under the assumption that m≥3m≥3 and n≥2mn≥2m. Let Uq(sl2)Uq(sl2) be the quantum enveloping algebra of sl2sl2 and let ⊠q⊠q be the q  -tetrahedron algebra. We first obtain an action of Uq(sl2)Uq(sl2) on the standard module of Jq(n,m)Jq(n,m). Then we display a CC-algebra homomorphism ϑ:Uq(sl2)→Tϑ:Uq(sl2)→T and show that T is generated by the image of ϑ and some central elements of T  . As an application, we also display an action of ⊠q⊠q on the standard module of Jq(n,m)Jq(n,m). These results are obtained by using the theory of Leonard pairs.