Feigin's map revisited

The aim of this note is to understand the injectivity of Feigin's map Fw by representation theory of quivers, where 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-known 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.