# Why are root systems presented in this confusing way?

I quote Bjorner and Brenti, "Combinatorics of Coxeter Groups."

We begin with a simple geometric lemma. Let $m \geq 3$ be an integer, let $\gamma = \pi/m$, and let $k, k'$ be real numbers such that $k k' = 4 \cos^2 \gamma$. Choose basis vectors $\beta$ and $\beta'$ in the Euclidean plane such that the angle between $\beta$ and $\beta'$ equals $\gamma$ and their lengths are related by

$|\beta'| = \frac{2 \cos \gamma}{k} |\beta|$

$|\beta| = \frac{2 \cos \gamma}{k'} |\beta'|$.

Let $r$ (resp $r'$) denote the orthogonal reflection in the line spanned by $\beta$ (resp $\beta'$).

Lemma

The coordinates $(q, q')$ of a point $q\beta + q'\beta'$ are transformed as follows by the orthogonal reflections:

$r'(q, q') = (-q, q' + kq)$

$r(q, q') = (q + k' q', -q')$

This seems crazy to me -- we define a variable $m$ which is never used again and place unnecessary restrictions on it; pull expressions for $k$ and $k'$ out of nowhere, only to have them turn out to be exactly what we need later. Not only that, but we don't even find out why we're defining all these variables in this way until much farther down the page. If I were to write this out, I would do something like this:

Let $\beta$ and $\beta'$ be linearly independent vectors in $\mathbb{R}^2$ which form an acute angle. Let $r$ (resp $r'$) denote orthogonal reflection about $\beta$ (resp $\beta'$). Then it is easy to see that

$r \beta' + \beta' = (2|\beta'| \cos \gamma)\, \hat\beta$,

and similarly

$r' \beta + \beta = (2 |\beta| \cos (-\gamma))\, \hat\beta'$.

Then we could define $k$ and $k'$ and write out matrices for $r$ and $r'$ in this basis. This isn't the only place I've seen this sort of presentation, either, so I'm guessing there must be some reason behind it that I'm missing. Can anyone explain why this is presented this way?

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$m$ appears in the standard presentation of a Coxeter group, so it's always lurking in the background even if it isn't being explicitly used. – Qiaochu Yuan Sep 11 '11 at 19:59
Yeah, I know that when we apply the lemma we'll have $\gamma = \pi/m$ in particular, but it's not used in the lemma in any way. – Daniel McLaury Sep 11 '11 at 20:07
I'm not familiar with the book... But it is not unusual that books be written with "mathematical adults" as intended audience, who should be more or less able to translate themselves between your two options. In any case, $m$ is an important parameter in that context—even if formally the reader will only find out, in principle, only later. It is a safe assumption that things will become clearer as one reads on—and a completely motivated exposition—one which doesn't ask the reader to suspend the kind of evaluation you are making until he has advanced more in the text—is rarely worth the effort. – Mariano Suárez-Alvarez Sep 11 '11 at 21:00