  We create a framework for studying symmetric chain decompositions of families
of finite posets based on the geometry of polytopes. Our framework unifies
almost all known results regarding symmetric chain decompositions of the Young
posets $L(m,n)$ --- arising as cells in the Bruhat decomposition of quotients
of $SL_{m+n+1}$ --- and yields unexpected new results. The methods we provide
are geometric in nature, systematic, and totally amenable to human analysis.
This allows us to discover new phenomena which are impenetrable to casework and
brute force computer search. In particular, our method yields perfect and near
perfect decompositions of various families of posets, which are intractable by
known methods. A fundamental tool we use is geometrical projection, which in
our framework cleanly unifies many different types of induction; as we move a
point from which we project between faces of our polytope, we alter the type of
induction. Moreover, projection allows us to decrease dimension and therefore
obtain a clear geometric intuition. We also provide additional tools for
producing decompositions, and discuss how the various decompositions behave
under products.
