  In this paper we investigate certain fusion relations associated to an
integrable vertex model on the square lattice which is invariant under $Sp(4)$
symmetry. We establish a set of functional relations which include a transfer
matrix inversion identity. The solution of these relations in the thermodynamic
limit allows us to compute the partition function per site of the fundamental
$Sp(4)$ representation of the vertex model. As a byproduct we also obtain the
partition function per site of a vertex model mixing the four and five
dimensional representations of the $Sp(4)$ symmetry.
