  The Kronecker product of two Schur functions $s_{\mu}$ and $s_{\nu}$, denoted
by $s_{\mu}*s_{\nu}$, is the Frobenius characteristic of the tensor product of
the irreducible representations of the symmetric group corresponding to the
partitions $\mu$ and $\nu$. The coefficient of $s_{\lambda}$ in this product is
denoted by $\gamma^{\lambda}_{{\mu}{\nu}}$, and corresponds to the multiplicity
of the irreducible character $\chi^{\lambda}$ in $\chi^{\mu}\chi^{\nu}.$
  We use Sergeev's Formula for a Schur function of a difference of two
alphabets and the comultiplication expansion for $s_{\lambda}[XY]$ to find
closed formulas for the Kronecker coefficients $\gamma^{\lambda}_{{\mu}{\nu}}$
when $\lambda$ is an arbitrary shape and $\mu$ and $\nu$ are hook shapes or
two-row shapes.
  Remmel \cite{Re1, Re2} and Remmel and Whitehead \cite{Re-Wh} derived some
closed formulas for the Kronecker product of Schur functions indexed by two-row
shapes or hook shapes using a different approach. We believe that the approach
of this paper is more natural. The formulas obtained are simpler and reflect
the symmetry of the Kronecker product.
