Let $\Delta$ be an irreducible reduced Dynkin diagram and let $\Pi\subseteq \Delta$ be a system of simple roots. I am interested in the representation theory of the group generated by elements $ \gamma_\alpha$, for $\alpha\in \Pi$, satisfying the following relations:
$\gamma_\alpha^4=e$,
$\gamma_\alpha^2=\gamma_\beta^2$, for all $\alpha,\beta\in \Pi$,
$\gamma_\alpha\gamma_\beta\gamma_\alpha^{-1}=\gamma_\beta$ if $\beta(\check{\alpha})$ is even,
$\gamma_\alpha\gamma_\beta\gamma_\alpha^{-1}=\gamma_\beta^{-1}$ if $\beta(\check{\alpha})$ is odd.
This group has $2^{p+1}$ elements, for $p:=|\Pi|$, since it is a central extension of $\mathbb{Z}_2^p$ by $\mathbb{Z}_2$.
In particular, there are $2^p$ one-dimensional representations and I'm wondering if there always exist irreducible representations in $2^{\lfloor p/2\rfloor}$ dimensions. If $\Delta$ is of type $A_{n-1}$, for $n\geq 3$, the group above is the inverse image of the diagonal subgroup in $\text{SO}(n)$ under the two-fold covering $\text{Spin}(n)\to \text{SO}(n)$ and the spin representations restricted to this subgroup are irreducible. I'm trying to see if this generalizes and would welcome any suggestions for references as well.
I think I understood the main points. First, note that if the Dynkin diagram has two vertices joined by two edges the group above is the direct product of the subgroups associated to the two subgraphs one obtains by deleting the two edges joining the two vertices. Furthermore, the group associated to G_2 is the same as for A_2. Therefore, we can assume that $\Delta$ is simply laced.
If $\Delta$ is of type $A_n$, one sees rather well why only the one or the two spin representations occur depending on $n$ even or odd. Let $\gamma_1,\dots,\gamma_n$ be linear endomorphisms of a complex vector space $V$ that satisfy $\gamma_i^2=-\text{id}_V$ and $\gamma_i$ anti-commutes with $\gamma_{i-1}$ and $\gamma_{i+1}$ and commutes with all other $\gamma_j$. Then $\gamma_1$ is semisimple with eigenvalues $\pm i$ and $\gamma_3,\dots,\gamma_n$ leaves the eigenspaces invariant. On can calculate in two dimensions what it means that
$\gamma_1=\begin{pmatrix} i&0\\ 0&-i \end{pmatrix}$ anti-commutes with $\begin{pmatrix} a&b\\ c&d \end{pmatrix}$, namely $a=d=0$.
This implies that $\gamma_2$ maps the $\pm i$-eigenspace to the $\mp$-eigenspace which proves that they have the same dimension. Iterating this one arrives at the conclusion that $\text{dim} V\geq 2^{\lfloor n/2\rfloor}$. Depending on $n$ even or odd, there are now either one or two irreducible representations with this dimension.
For $\Delta$ of type $D_4$, there is one basis root, let's call it $\alpha_1$, with three outgoing edges and the other basis roots $\alpha_2,\alpha_3,\alpha_4$ are only connected to this one. There are four (= 4 choose three) irreducible representations of dimension 2 which can be implemented by
$\gamma_1=\begin{pmatrix} i&0\\ 0&-i \end{pmatrix}$ and $\{\gamma_2,\gamma_3,\gamma_4\}\subseteq \left\{\pm \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix},\pm \begin{pmatrix} 0&i\\ i&0 \end{pmatrix}\right\}$.
The cases, of $\Delta$ of type $E_6$, $E_7$ and $E_8$, can then be obtained as a combination of these.