Feigin's map revisited

TL;DR

This paper investigates the injectivity of Feigin's map using quiver representation theory and Ringel-Hall algebras, also extending Reineke's monomial bases to non-simply-laced cases.

Contribution

It provides a new understanding of Feigin's map injectivity via representation theory and generalizes monomial bases to broader types.

Findings

Established the injectivity of Feigin's map using quiver representations.

Generalized Reineke's monomial bases to non-simply-laced cases.

Connected algebraic and representation-theoretic approaches for finite type quivers.

Abstract

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.