Proof of a theorem on Reflection Groups

Question

I am reading the book Finite Reflection Groups by Grove and Benson.

I didn't understand the following proof. See $(a_1,t)$. What is $t$ here? Then Why the inequality $(r,t)-2(r,r_{i_1})(r_{i_1},t)<(r,t)?$ We know that $(r,r_{i_1})>0$, but it can happen that $(r_{i_1},t)<0$.

This means, the $t$ should be specified properly; is this correct?

OK, once we obtain $(a_1,t)<(r,t)$, and if $a_1\notin\Pi$, then we obtain $r_{i_2}\in \Pi$ and $a_2$ with $(a_2,t)<(a_1,t)$. Question is, here also, is $t$ arbitrary or it is chosen initially? I mean, does such $t$ exists?

Answer 1

Apparently $(t,r_i)>0$ for all the simple roots $r_i$. In other words $t$ is some vector in the fundamental Weyl chamber. May be $t$ is Grover&Benson's notation for the half-sum of positive roots? May be it is just a random vector in that chamber. After all, such a vector is selected so that using it we can define

• the positive roots as roots $r$ such that $(r,t)>0$, and
• the simple roots as those positive roots that cannot be written as a sum of two positive roots.

Anyway, the induction then works as described.