  We study the Jacobi-Trudi-type determinant which is conjectured to be the
q-character of a certain, in many cases irreducible, finite-dimensional
representation of the quantum affine algebra of type D_n. Unlike the A_n and
B_n cases, a simple application of the Gessel-Viennot path method does not
yield a positive sum expression of the determinant over a set of tuples of
paths. However, applying an additional involution and a deformation of paths,
we obtain a positive sum expression over a set of tuples of paths, which is
naturally translated into the one over a set of tableaux on a skew diagram.
