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The Coxeter number is the order of a Coxeter element in a finite irreducible Coxeter group.

The finite representation type of a quiver condition means there are only finitely many isomorphism classes of indecomposable finite-dimensional $kQ$-modules. Equivalently, every finite-dimensional module is a direct sum of members of a finite list of indecomposables.

For the $E_6$ case, we give the reflection argument underlying the Gabriel theorem, rather than using its conclusion as the proof. Since the underlying graph is a tree, every orientation is acyclic. At a sink $i$, writefor the sum of the incoming arrow maps. If an indecomposable $V$ is not the simple $S_i$, then $h$ is surjective: a nonzero complementary subspace to its image at the sink would be a direct summand supported only at $i$. The Bernstein–Gelfand–Ponomarev reflection functor reverses the arrows and replaces $V_i$ by $\ker h$, with reversed maps the component projections. The source reflection replaces that kernel by the cokernel of its inclusion into $⨁V_j$, recovering $V_i$. The inclusion is injective, so the reflected representation has no direct summand $S_i$ at the new source. If it decomposed, applying the inverse would decompose $V$. Thus reflection preserves indecomposability except that it annihilates $S_i$.

On a dimension vector of a quiver representation, it acts by the integral simple reflectionLet $B=2I-\mathrm{Adj}(E_6)$. These reflections preserve the positive-definite quadratic form $d^{\mathsf{T}}Bd$. Positivity can be checked without a classification theorem: label the graph by the chain $1-2-3-4-5$ and attach vertex $6$ to $3$. The leading principal minors are $2,3,4,5,6,3$, all positive.

Choose a topological ordering so arrows point from smaller to larger indices. Reflect successively at sinks $6,5,\dots ,1$. This restores the orientation after one full cycle. The transformation of dimension vectors is the Coxeter element $C=s_1{s}_2\cdots s_6$. If $U$ has diagonal entries one and entry $-1$ above the diagonal at an arrow, direct multiplication of the reflection matrices givesThus $C$ has no fixed vector. It is an integral isometry of a positive definite form, so it has finite order $h$: the image of each lattice basis vector lies in a finite set of vectors of the same norm, giving only finitely many possible isometries. Since $C^h=I$ and $I-C$ is invertible,If an indecomposable survived all $h$ reflection cycles, its dimension vectors $d,Cd,\dots ,C^{h-1}d$ would all be nonzero and componentwise nonnegative, contradicting their zero sum. Therefore it reaches a simple at one of the finitely many reflection steps and is then annihilated. Applying the inverse reflections uniquely reconstructs it from that simple. There are at most $6h$ such stopping positions, provingThis supplies the requested finite-type proof sketch.

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.

The Coxeter number is the order of a Coxeter element. In the stated families the values areHere $D_{2n}$ is the dihedral group of order $2n$, $S_n$ is the symmetric group, $S{S}_n$ is the signed symmetric group, and $ES{S}_n$ is the even signed symmetric group.

Let $B$ be the matrix whose $i$th column consists of the coordinates of ${\beta }_i$ in the basis $(e_j)$. Define the upper-triangular matrix $U$ and lower-triangular matrix $L$ byThe first identity in part b says $I=BU$, so $B=U^{-1}$. The second says that the matrix $C$ of $\sigma (c)$ is $C=BL=U^{-1}L$. Since $U$ has diagonal entries one, $\det U=1$, and thereforeThus $\det (tU-L)$ is the characteristic polynomial of the Coxeter element in its geometric representation.

At $t=1$, the diagonal entries of $U-L$ are $2$, while every off-diagonal $(i,j)$ entry isHence $U-L$ is exactly the Coxeter Gram matrix $G(W)$, and part c givesFor a Finite Coxeter group the Gram matrix is positive definite, and for a Hyperbolic Coxeter group it is nondegenerate with Lorentzian signature. In either case $\det (I-C)\ne 0$, so $1$ is not an eigenvalue of $C$ and the Coxeter element fixes no nonzero vector.

For an Affine Coxeter group, the Gram form has a nonzero radical. If $0\ne v\in \mathrm{rad}G(W)$, then $⟨v,e_i⟩=0$ for every $i$, and every generating reflection satisfiesTheir product $\sigma (c)$ therefore fixes $v$. Thus every affine Coxeter element has a nonzero fixed vector.