I was wondering what is the group theoretic way to find the resulting charges of matter fields after a scalar field is given a vev.
In the case of the EW symmetry breaking, one can directly read the charges from the Lagrangian by setting the Higgs field $H=v+h'$ and going in the unitary gauge.
Given a gauge group $\mathcal{G}$, a set of field with their charges under that group. What is the way to find the charges if I give a vev to $n$ fields under the remnant group $\mathcal{G}_\text{br.}$. This is a priori totally unrelated to any Lagrangian and should have a purely group theoretic answer.
A simple example would be $\mathcal{G}=U(1)^k$ with $m$ fields. If I give a vev to $n$ of them, we'll have $U(1)^k\to U(1)^{k-n}$ (assuming the $n$ fields have linearly independent charges). My problem is that I can't find how to get the charges.
I would also be interested in the non-abelian case, and with not only scalar fields but other spin in the spectrum. Any references would also be very welcome!