= Solution
For a real-character irreducible molecular representation with <character> $\chi$, the multiplicity of the <fully symmetric representation> in its tensor square is
$$
m_{\rm sym}=\frac1{|G|}\sum_{g\in G}\chi(g)^2
=\frac1{|G|}\sum_{g\in G}|\chi(g)|^2
=\boxed{1},
$$
by <character orthogonality>. The identity uses the real-character assumption appropriate to the usual real <irreducible representations> intended here. For a general complex <irreducible representation>, the <trivial representation in a tensor square> has multiplicity one precisely when it is self-dual, and zero otherwise, by <Schur's lemma>. The universally valid tensor product is $V\otimes V^*$.
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