The terwilliger algebra of the twisted grassmann graph: The thin case

The Terwilliger algebra T (x) of a finite connected simple graph Γ with respect to a vertex x is the complex semisimple matrix algebra generated by the adjacency matrix A of Γ and the diagonal matrices E_i^∗(x) = diag(v_i) (i = 0, 1, 2, … ), where v_i denotes the characteristic vector of the set of vertices at distance i from x. The twisted Grassmann graph J_q (2D +1, D) discovered by Van Dam and Koolen in 2005 has two orbits of the automorphism group on its vertex set, and it is known that one of the orbits has the property that T (x) is thin whenever x is chosen from it, i.e., every irreducible T (x)-module W satisfies dim E_i^∗(x)W ≤ 1 for all i. In this paper, we determine all the irreducible T (x)-modules of J_q (2D + 1, D) for this “thin” case.