The Terwilliger algebra of the twisted Grassmann graph: the thin case

The Terwilliger algebra T(x) of a finite connected simple graph Gamma with respect to a vertex x is the complex semisimple matrix algebra generated by the adjacency matrix A of Gamma 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 Grossmann graph (J) over tilde (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) over tilde (q) (2D + 1, D) for this thin case.