Potentially semistable deformation rings for representations with values in $\mathrm{PGL}_n$

We study the potentially semistable deformation rings for Galois representations taking their values in ${\rm PGL}_n$, by comparing them to the deformation rings for ${\rm GL}_n$. As an application, we state an analogue of the Breuil–Mézard conjecture, and we show that the case of ${\rm PGL}_n$ follows from the case of ${\rm GL}_n$.