  The aim of this note is to understand the injectivity of Feigin's map
$\mathbf{F_w}$ by representation theory of quivers, where $\mathbf{w}$ is the
word of a reduced expression of the longest element of a finite Weyl group.
This is achieved by the Ringel-Hall algebra approach and a careful studying of
a well-knwon total order on the category of finite-dimensional representations
of a valued quiver of finite type.
  As a byproduct, we also generalize Reineke's construction of monomial bases
to non-simply-laced cases.
