Let $\rho \colon {\mathrm{Gal }} \to {\mathrm{GL}_2}({\Bbb F}_p)$ be a galois representation.
Q. Why does the equality ${\mathrm{ad}}^{0}(\rho) = {\mathrm{Sym}}^2(\rho)$ hold?
Adjoint-$0$ representation is the action by conjunction on trace $0$ matrices in ${\mathrm{End}}(V)$, whereas the symmetric tensor is the sub-vector space in $V \otimes V$ invariant under permutation of indices.