Exercise 4, § 6

Lie Groups and Lie Algebras, V

With the assumptions and notation of no. 1, denote by $n_{ij}$ the order of $s_{H_i} s_{H_j}$ , and put $a_{ij} = -2 \cos \frac{\pi}{n_{ij}}$ . Then $a_{ii} = 2$ and $a_{ij} \leq 0$ if $i \neq j$ , cf. § 3. Show $^{12}$ , by using the method of the preceding exercise, that

\begin{vmatrix} X & a_{12} & \ldots & a_{1l} \\ a_{21} & X & \ldots & a_{2l} \\ \vdots & \vdots & \ddots & \vdots \\ a_{l1} & a_{l2} & \ldots & X \end{vmatrix} = \prod_{i=1}^{l} (X - 2 \cos(\pi m_i / h)),

where $m_1, \ldots, m_l$ are the exponents of W.

$^{12}$ For more details, see: H. S. M. COXETER, The product of the generators of a finite group generated by reflections, Duke Math. Journal., v. XVIII (1951), p. 765-782.

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