Some background settings

Let $V$ be a $n$-dimensinal vector space over $\mathbb{R}$, $V^\ast$ be its dual space. A reflection transformation $r$ on $V$ is a linear transformation that fixes a hyperplane $H$ and maps a vector $\alpha^\vee$ to $-\alpha^\vee$. Or in other words, there is a linear functional $\alpha\in V^\ast$, a vector $\alpha^\vee\in V$, $\langle\alpha, \alpha^\vee\rangle=2$ so that $$r(v) = v - \langle \alpha, v\rangle \alpha^\vee,\quad v\in V.$$

Now let $A=(a_{ij})_{1\leq i,j\leq n}$ be a real matrix which satisfy $a_{ii}=2$ for all $1\leq i\leq n$, choose a basis $\{\alpha_i, 1\leq i\leq n\}$ in $V^\ast$, and let $\{\alpha_i^\vee, 1\leq i\leq n\}$ be vectors in $V$ so that $$\langle \alpha_i, \alpha_j^\vee\rangle = a_{ij}.$$ and $r_i$ be the reflection $$r_i(v) = v -\langle \alpha_i, v\rangle \alpha_i^\vee.$$ $W$ be the reflection group generated by $\{r_i,i=1,\ldots, n\}$.

Question 1: Is $W$ a Coxeter group or not? If not, what restrictions must be put on the matrix $A$ to make $W$ a Coxeter group?

For such a matrix $A$, $A$ is called the Cartan matrix of the Coxeter group $W$. $A$ gives a geometric realization of $W$ as a set of reflections.

Question 2: in the study of a Coxeter group and its geometric realizations, there is often a convex cone $C$, called the Tit's cone, on which the group acts discretely. Not all reflection groups preserve such a convex cone. For example when $A$ is a 2x2 matrix and the entries satisfy $a_{12}a_{21}<0$. (see Vinberg's paper) So if $A$ defines a Coxeter group as in question 1, then what further restrictios on $A$ must be put to make sure this reflection group preserves something like Tit's cone?

Why I have this problem

I want to have an example which is a reflection group (generated by reflections about hyperplanes) but not a Coxeter group (i.e. not isomorhpic to any abstract Coxeter group which is defined by a Coxeter matrix). The wiki says reflection groups are also Coxeter groups, but as I could remember if the toplogy of the hyperplanes are not well behaved then the reflection group in this case (must be an infinite group) will not be a Coxeter group. I can't recall where I saw this.

Also I want to know better how the requirements on a Cartan matrix for Coxeter groups/Kac-Moody algebras determines the geometry of the root system. For example, in the definition of a generalized Cartan matrix, what happens if we allow the non-diagonal entries can be positive?

Please let me know if you think my problem is not clearly formulated.


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