22 Polytope Projects
The set P
n
={x: ρ(x) = n
∈
Z} are called levels of G.
G can be regarded either as oriented or non-oriented graph. The
accessibility relation in an oriented graph defi nes a partial order on P. If
there are no multiple arcs, G is the Hasse diagram of this poset. The paths
in non-oriented graphs are the Hasse walks.
As for the differential posets the “down” and “up” operators can be
defi ned.
Let G= (P, ρ, E) be a graded graph. Linear operators U and D are defi ned
by:
∑
∈
=
E)y,x(
y)y,x(mUx
;
∑
∈
=
E)y,x(
x)y,x(mDy
(2.5)
Here m (x, y) is the multiplicity of the edge (x, y) in E.
The main idea of Fomin was to consider the pairs of graded graphs
G
1
= (P, ρ, E
1
) and G
2
= (P, ρ, E
2
) with a common set of vertices and a common
rank function.
The oriented ...