Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 102 / 5 / Solution

A simply-laced positive-definite Coxeter graph has no multiple edges and has positive-definite symmetric Cartan matrix, with diagonal entries $2$ and entry $-1$ precisely across an edge. Its connected components are exactly the ADE classification

The graph underlying $1\to 2$ is $A_2$. For simple roots ${\alpha }_1,{\alpha }_2$ with $({\alpha }_i,{\alpha }_i)=2$ and $({\alpha }_1,{\alpha }_2)=-1$, its roots areThere is one base for each Weyl chamber, hence six bases. The Weyl group is generated by the two root reflections withso $W\simeq S_3$. Relative to a chosen base, the two Coxeter elements are $s_1{s}_2$ and $s_2{s}_1$; both have order three.

A representation of the quiver is one linear map $f:V_1\to V_2$. Choosing bases that put $f$ in rank normal form decomposes it into copies ofThese are indecomposable and have dimension vectors $(1,0)$, $(0,1)$, and $(1,1)$, the three positive roots of $A_2$. More generally, Gabriel theorem says that for any orientation of a simply-laced positive-definite Dynkin graph, indecomposable quiver representations are in bijection with its positive roots. The finite ADE root system therefore gives finitely many indecomposables.