  We consider four examples of short step lattice paths confined to the quarter
plane. These are the Kreweras, Reverse Kreweras, Gessel, and Mishna-Rechnitzer
lattice paths.The Reverse Kreweras are straightforward to solve and thus
interesting as a contrast to the Kreweras paths and Gessel paths as the latter
two have historically been significantly more difficult to solve. The
Mishna-Rechnitzer paths are interesting as they are associated with an infinite
order group. We will give some geometrical insight into all these properties by
considering the Weyl chambers associated with their step sets.For Reverse
Kreweras paths the Weyl chamber walls coincide with the quarter plane boundary
and hence the problem is readily solvable by Bethe Ansatz or by using the
Gessel-Zeilberger Theorem. For Kreweras paths the quarter plane corresponds to
the union of two adjacent Weyl Chambers and hence neither the Bethe Ansatz nor
the Gessel-Zeilberger Theorem are directly applicable making the problem
considerably more difficult to solve. Similarly, the quarter plane for Gessel
paths is the union of three Weyl chambers. For Mishna-Rechnitzer paths the step
set has non-zero barycenter leading to an affine dihedral reflection group. The
affine structure corresponds to the drift in the random walk. The quarter plane
is the union of an infinite number of Weyl alcoves.
