  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
$E_i^*(x)=\operatorname{diag}(v_i)$ $(i=0,1,2,\dots)$, where $v_i$ denotes the
characteristic vector of the set of vertices at distance $i$ from $x$. The
twisted Grassmann graph $\tilde{J}_q(2D+1,D)$ 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 $\dim
E_i^*(x)W\leqslant 1$ for all $i$. In this paper, we determine all the
irreducible $T(x)$-modules of $\tilde{J}_q(2D+1,D)$ for this "thin" case.

Combinatorics