  The long mean free path of $K^+$ mesons in nuclear matter makes this particle
a suitable messenger for the dynamics of nucleus-nucleus reactions at
intermediate energies (100 MeV to 3 GeV per nucleon). A prerequisite for this
is the knowledge of the elementary production cross sections $\pi N \rightarrow
\Sigma K$. Here these cross sections are studied for the first time with the
explicite inclusion of the relevant baryon resonances up to 2 GeV as
intermediate states. The baryon resonances -- $N(1710)\, I(J^P) = \frac{1}{2}
(\frac{1}{2}^+),\, N(1720)\, \frac{1}{2} (\frac{3}{2}^+)$ and $\Delta(1920)\,
\frac{3}{2} (\frac{3}{2}^+)\,$ -- are taken into account coherently in the
calculations of the $\pi N \rightarrow \Sigma K$ process. (We refer to this
model as the `resonance model'.) Also $K^*(892)\frac{1}{2} (1^-)$ vector meson
exchange is included. It is shown that the total cross sections for different
channels of the $\pi N \rightarrow \Sigma k$ reactions, i.e. $\pi^+ p
\rightarrow \Sigma^+ K^+$, $\pi^- p \rightarrow \Sigma^- K^+$, $\pi^+ n
\rightarrow \Sigma^0 K^+$ ($\pi^- p \rightarrow \Sigma^- K^+$) and $\pi^0 p
\rightarrow \Sigma^0 K^+$ differ not only by absolute values but also by their
energy dependence. This shape differences are due to the mixture of the isospin
$I = 3/2$ $\Delta(1920)$ with isospin $I = 1/2$ nucleon resonances. However,
this $I = 3/2$ resonance does not give a contribution to the $\pi N \rightarrow
\Lambda K$ reactions. So the shapes of the total cross sections $\pi N
\rightarrow \Lambda K$ for different isospin projections are the same. In spite
of this, such cross sections averaged over different isospin projections in the
same multiplet
