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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.
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The point of your question is: What is the relation between the two representations you describe of a Coxeter group?

To each Coxeter group, one can associate a Cartan matrix in an even more general sense, where you allow real entries but require that $a_{ij}a_{ji}=4\cos^2\left(\frac{\pi}{m(i,j)}\right)$, where the $m(i,j)$ are the data for the Coxeter system. (I'm taking $S=\{1,\ldots,n\}$.) Then the reflections $R_i$ that you describe are reflections generating a group isomorphic to the Coxeter group. The reflections $\sigma$ you describe are these reflections, in the case where the (generalized generalized) Cartan matrix is chosen to be symmetric.

In some cases (called "crystallographic"), the Coxeter system admits an integer (and symmetrizable) Cartan matrix. In this case, we get a representation of the Coxeter group as a group of isometries of a symmetric bilinear form derived from the Cartan matrix.

Anyway, once this is understood, the question about the Coxeter element is also answered (unless I misunderstood the question).

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  • $\begingroup$ Thanks Nathan for your answer. I clarified my question above (see the "added later" paragraph at the bottom) to indicate more directly what I want to know. I'm not quite sure if your question answers this. - "The reflections $\sigma$ you describe": You mean $\sigma(i)$, right? - I guess the Cartan matrix you associate to a Coxeter system is just the matrix $a_{ij} = - 2\cos \left(\frac{\pi}{m(i,j)} \right)$? - I don't really understand what you mean with "but require that $a_{ij}a_{ji} = ...$", since this requirement makes no sense if we don't have a Coxeter system at hand $\endgroup$ – Leon Lang Aug 27 '17 at 10:16
  • $\begingroup$ I'm starting with a Coxeter group and constructing (with lots of freedom to make choices) a Cartan matrix (in this much more general sense). Based on your addition to the question, you just want to reverse the process. Assuming that we have numbered $S$ as $\{s_1,\ldots,s_n\}$, you just define $m(s_i,s_j)$ so that $a_{ij}a_{ji}$ is $4\cos^2\left(\frac{\pi}{m(s_i,s_j)}\right)$. $\endgroup$ – Nathan Reading Aug 28 '17 at 19:19
  • $\begingroup$ A few more comments: First, since you are starting with a symmetric Cartan matrix, you can just define $m(s_i,s_j)$ so that $a_{ij}=2\cos\frac{\pi}{m(s_i,s_j)}$. $\endgroup$ – Nathan Reading Aug 28 '17 at 19:29
  • $\begingroup$ Second, you'll see that the only possibilities for $m(s_i,s_j)$ are $2$, $3$, and $\infty$, corresponding to Cartan matrix entries $0$, $-1$, and $-2$. If you have entries $a_{ij}<-2$, then you can define the right Coxeter group by still taking $m(s_i,s_j)=\infty$, but then the representation you get from defining your $\sigma(s_i)$ reflections is not the same as the representation you get from the $R_i$. There's nothing you can do about that, and I assume the spectra of the Coxeter elements also don't agree, although I've not checked. (You can work this out in the $n=2$ case and see.) $\endgroup$ – Nathan Reading Aug 28 '17 at 19:35
  • $\begingroup$ Ah, I see. That probably means that A'Campos article is not cited in all the papers because his result is applicable but because his proof works also for Coxeter transformations in terms of generalized Cartan matrices. As soon as I will have checked this I will probably write another answer. $\endgroup$ – Leon Lang Aug 28 '17 at 21:53

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