  We study the basic features of BPS quiver mutations in 4D $\mathcal{N}=2$
supersymmetric quantum field theory with $G=ADE$ gauge symmetries.\ We show,
for these gauge symmetries, that there is an isotropy group
$\mathcal{G}_{Mut}^{G}$ associated to a set of quiver mutations capturing
information about the BPS spectra. In the strong coupling limit, it is shown
that BPS chambers correspond to finite and closed groupoid orbits with an
isotropy symmetry group $\mathcal{G}_{strong}^{G}$ isomorphic to the discrete
dihedral groups $Dih_{2h_{G}}$ contained in Coxeter$(G) $ with $% h_{G}$ the
Coxeter number of G. These isotropy symmetries allow to determine the BPS
spectrum of the strong coupling chamber; and give another way to count the
total number of BPS and anti-BPS states of $\mathcal{N}=2$ gauge theories. We
also build the matrix realization of these mutation groups $%
\mathcal{G}_{strong}^{G}$ from which we read directly the electric-magnetic
charges of the BPS and anti-BPS states of $\mathcal{N}=2$ QFT$_{4}$ as well as
their matrix intersections. We study as well the quiver mutation symmetries in
the weak coupling limit and give their links with infinite Coxeter groups. We
show amongst others that $\mathcal{G}_{weak}^{su_{2}}$ is contained in
${GL}({2,}\mathbb{Z}) $; and isomorphic to the infinite Coxeter
${I_{2}^{\infty}}$. Other issues such as building
$\mathcal{G}%_{weak}^{so_{4}}$ and $\mathcal{G}_{weak}^{su_{3}}$ are also
studied.
