Feigin's map revisited

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.