  We present a necessary and sufficient condition for a finite-dimensional
highest weight representation of the $sl_2$ loop algebra to be irreducible. In
particular, for a highest weight representation with degenerate parameters of
the highest weight, we can explicitly determine whether it is irreducible or
not. We also present an algorithm for constructing finite-dimensional highest
weight representations with a given highest weight. We give a conjecture that
all the highest weight representations with the same highest weight can be
constructed by the algorithm. For some examples we show the conjecture
explicitly. The result should be useful in analyzing the spectra of integrable
lattice models related to roots of unity representations of quantum groups, in
particular, the spectral degeneracy of the XXZ spin chain at roots of unity
associated with the $sl_2$ loop algebra.
