Fixed-point subspace of a group action ($\mathrm{Fix}(H)=\{v:hv=v\forall h\in H\}$)

For a linear group representation, the common fixed vectors of a subgroup form an invariant subspace for any equivariant map restricted to it. Its dimension is central to symmetry-based branching theorems. It need not be invariant under the entire group; conjugate subgroups have conjugate fixed-point subspaces.