This question is related to the following question about Coxeter transformations that I asked and recently answered myself. For completeness I also write full definitions in the new question.

**The Coxeter transformation of a Coxeter system:**

Let $(W,S)$ be a Coxeter system, i.e. $W$ is a group, $S \subset W$ is a finite set and there are numbers $m(s,s') \in \mathbb{N}_{\geq 1} \cup \{\infty\}$ (with $m(s,s) = 1$ and $m(s,s') \geq 2$ for $s \neq s'$) such that $W$ has a presentation with generators $S$ and relations $(ss')^{m(s,s')}$ for all $s,s' \in S$ with $m(s,s') \neq \infty$. Then the *canonical representation* of $(W,S)$ is the group homomorphism $\sigma: W \to \text{GL}\left(\mathbb{R}^S\right)$ such that $\sigma$ is given on $S$ by

$$\sigma(s)(e(s')) = e(s') + 2 \text{cos} \left( \frac{\pi}{m(s,s')}\right)\cdot e(s).$$

Then for a total order $S = \{s_1, \dots, s_n\}$, the *Coxeter element* of $(W,S)$ is defined as the product $c = s_1 \cdots s_n$ and the corresponding *Coxeter transformation* is $C = \sigma(c) \in \text{GL}\left( \mathbb{R}^S\right)$. In the case that the Coxeter graph of $(W,S)$ (i.e. the graph with vertex set $S$ and an edge between $s,s'$ labeled with $m(s,s')$ whenever $m(s,s') \geq 3$) is a tree, the Coxeter elements of different total orders are all conjugate.

**The Coxeter transformation of a generalized Cartan matrix**:

In the paper The spectral radius of the Coxeter transformations for a generalized Cartan matrix, Claus Ringel defines a Coxeter transformation as follows, in several steps:

A *generalized Cartan matrix* of size $n$ is a matrix $A \in M^{n \times n}(\mathbb{Z})$ such that for all $i \neq j$ the following properties are satisfied:

- $A_{ii} = 2$
- $A_{ij} \leq 0$
- $A_{ij} \neq 0 \Leftrightarrow A_{ji} \neq 0$

He then goes on to define the reflection $R_i: \mathbb{R}^n \to \mathbb{R}^n$ as the linear map (depending on $A$) which is given on the canonical basis by $e(j) \mapsto e(j) - A_{ji}e(i)$.

Now if $\pi: \{1, \dots, n\} \to \{1, \dots, n\}$ is any permutation, he calls the product $C(A, \pi) := R_{\pi(n)} \cdots R_{\pi(1)}$ a *Coxeter transformation* for $A$.

**Question:**

What precisely is the relation between those two notions of Coxeter transformations (and how do properties of Coxeter systems and properties of generalized Cartan matrices translate via this relation)?

**Context:**

I'm trying to understand the asymptotic behaviour of Coxeter transformations in the context of path algebras of wild quivers. All papers essentially cite the paper from Claus Ringel from above and a paper from Norbert A'Campo, in which the two definitions given in this question appear. I already made a link between the Coxeter transformations of path algebras and Coxeter transformations of generalized Cartan matrices in the last question and now want to understand the link between the remaining two definitions in this question.

*Added later:* I'd like to ask a more low-level question in order to indicate more directly what I want to understand. In fact, I'm only interested in *symmetric* generalized Cartan matrices (GCMs), since those are the GCMs that arise in the context of finite-dimensional quiver algebras (see my other question I linked at the top). I want to have a way how to associate to every symmetric GCM A (together with a permutation $\pi: \{1, \dots, n\} \to \{1, \dots, n\}$) a Coxeter system $(W,S)$ (together with an ordering $S = \{s_1, \dots, s_n\}$) such that the graphs associated to $A$ and $(W,S)$ coincide (or share similar properties) and such that the following holds:

- Let $C(A, \pi)$ be the Coxeter transformation of $A$ and $\sigma(c)$ be the corresponding Coxeter transformation of $(W,S)$. Then the spectra of eigenvalues of the two coincide, or at least the spectral radii coincide.