as in Example \[ExEps\](d)

-   ${ {\mathbb Z} }_2^{\varepsilon}=({ {\mathbb Z} }_2*{ {\mathbb Z} }_2)\times({ {\mathbb Z} }_2*{ {\mathbb Z} }_2)$ for ${\varepsilon}$ as in Example \[ExEps\](e)

-   For ${\varepsilon}$ as in Example \[ExEps\](f), we cannot write ${ {\mathbb Z} }_2^{\varepsilon}$ as an iteration of free and direct products.

The full group $C^*$-algebra associated to $\mathbb Z_2^{\varepsilon}$ is the following universal $C^*$-algebra: $$C^*(\mathbb Z_2^{\varepsilon})=C^*(z_1,\ldots, z_n\;|\; z_i=z_i^*, z_i^2=1, z_iz_j=z_jz_i \textnormal{ if } {\varepsilon}_{ij}=1)$$

\[RepSphere\] Let $H$ be a two-dimensional Hilbert space with orthonormal basis $e_1$ and $e_2$.

-   For $n=2$ and ${\varepsilon}={\varepsilon}_{\operatorname{free}}$, the Coxeter group $\mathbb Z_2^{\varepsilon}=\mathbb Z_2*\mathbb Z_2$ may be represented on $H$ by $\pi:C^*(\mathbb Z_2*\mathbb Z_2)\to B(H)$ defined as: $$z_1\mapsto a:=\begin{pmatrix}-1&0\\0&1\end{pmatrix},\qquad
    z_2\mapsto b:=\begin{pmatrix}0&1\\1&0\end{pmatrix}$$ Note that $a$ and $b$ do not commute.

-   Let $n\in{ {\mathbb N} }$ and ${\varepsilon}$ be arbitrary. Put $H_{ij}:=H$ for $1\leq i<j\leq n$. The representation $\sigma_{{\varepsilon}}: C(\mathbb Z_2^{\varepsilon})\to B(\bigoplus_{i<j} H_{ij})$ given by $$\sigma_{{\varepsilon}}(z_k)_{|H_{ij}}=\begin{cases} 
    a &\textnormal{ if } k=i\textnormal{ and }{\varepsilon}_{ij}=0\\
    b &\textnormal{ if } k=j\textnormal{ and }{\varepsilon}_{ij}=0\\
    \operatorname{id}_H & \textnormal{ otherwise }\end{cases}$$ is such that $\sigma_{{\varepsilon}}(z_i)$ and $\sigma_{{\varepsilon}}(z_j)$ commute if and only if ${\varepsilon}_{ij}=1$.

-   We may represent the ${\varepsilon}$-sphere on the Coxeter group $\mathbb Z_2^{\varepsilon}$ via: $${\varphi}_{\varepsilon}:C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})\to C^*(\mathbb Z_2^{\varepsilon}),\qquad x_i\mapsto \frac{1}{\sqrt 