  The normal form for an n-dimensional map with irreducible nilpotent linear
part is determined using sl2-representation theory. We sketch by example how
the reducible case can also be treated in an algorithmic manner. The
construction (and proof) of the sl2-triple from the nilpotent linear part is
more complicated than one would hope for, but once the abstract sl2 theory is
in place, both the description of the normal form and the computational
splitting to compute the generator of the coordinate transformation can be
handled explicitly in terms of the nilpotent linear part without the explicit
knowledge of the triple. If one wishes one can compute the normal form such
that it is guaranteed to lie in the kernel of an operator and one can be sure
that this is really a normal form with respect to the nilpotent linear part;
one can state that the normal form is in sl2-style. Although at first sight the
normal form theory for maps is more complicated than for vector fields in the
nilpotent case, it turns out that the final result is much better. Where in the
vector field case one runs into invariant theoretical problems when the
dimension gets larger if one wants to describe the general form of the normal
form, for maps we obtain results without any restrictions on the dimension. In
the literature only the 2-dimensional nilpotent case has been described sofar,
as far as we know.
