= Solution
At $t=1$, the diagonal entries of $U-L$ are $2$, while every off-diagonal $(i,j)$ entry is
$$
-a_{ij}=-2\cos\left(\frac{\pi}{m_{ij}}\right).
$$
Hence $U-L$ is exactly the <Coxeter Gram matrix> $G(W)$, and part c gives
$$
\det(I-C)=\det G(W).
$$
For a <Finite Coxeter group> the Gram matrix is <positive-definite matrix>[positive definite], and for a <Hyperbolic Coxeter group> it is nondegenerate with Lorentzian signature. In either case $\det(I-C)\ne0$, so $1$ is not an <eigenvalue> of $C$ and the <Coxeter element> fixes no nonzero vector.
For an <Affine Coxeter group>, the Gram form has a nonzero <radical of a bilinear form>[radical]. If $0\ne v\in\operatorname{rad}G(W)$, then $\langle v,e_i\rangle=0$ for every $i$, and every generating reflection satisfies
$$
\sigma(x_i)v=v-\langle v,e_i\rangle e_i=v.
$$
Their product $\sigma(c)$ therefore fixes $v$. Thus every affine Coxeter element has a nonzero fixed vector.
Solved by gpt-5.6-sol high.
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