  The spectral decomposition of the path space of the vertex model associated
to the level $l$ representation of the quantized affine algebra
$U_q(\hat{sl}_n)$ is studied. The spectrum and its degeneracy are parametrized
by skew Young diagrams and what we call nonmovable tableaux on them,
respectively. As a result we obtain the characters for the degeneracy of the
spectrum in terms of an alternating sum of skew Schur functions. Also studied
are new combinatorial descriptions (spectral decomposition) of the Kostka
numbers and the Kostka--Foulkes polynomials. As an application we give a new
proof of Nakayashiki--Yamada's theorem about the branching functions of the
level $l$ basic representation $l\Lambda_k$ of $\hat{sl}_n$ and a
generalization of the theorem.
