The Terwilliger algebras of Grassmann graphs

Let J(q)(n, m) denote the Grassmann graph with vertex set X and diameter min{m, n - m}. Fix a vertex x is an element of X. Let T = T(x) denote the Terwilliger algebra of J(q) (n, m) corresponding to x. In this paper we study the structure of T under the assumption that m >= 3 and n >= 2m. Let U-q(sl(2)) be the quantum enveloping algebra of sl(2) and let boxed times(q) be the q-tetrahedron algebra. We first obtain an action of U-q(sl(2)) on the standard module of J(q) (n, m). Then we display a C-algebra homomorphism : U-q(sl(2)) -> T and show that T is generated by the image of and some central elements of T. As an application, we also display an action of boxed times(q) on the standard module of a J(q)(n, m). These results are obtained by using the theory of Leonard pairs. (C) 2015 Elsevier Inc. All rights reserved.