S3-invariant Yukawa matrix for a doublet and a sign singlet ($\lambda =\mathrm{diag}(a,a,b)$)

Let the left and right fermions transform in $D\oplus \epsilon$, where $D$ is the two-dimensional irreducible representation of the symmetric group $S_3$ and $\epsilon$ is the sign representation. Let the scalar field be a trivial representation. In matching orthonormal bases, invariance of ${\bar{\psi }}_L\lambda {\psi }_R$ says that $\lambda$ commutes with $D\oplus \epsilon$. The Schur lemma forces each diagonal irreducible block to be scalar and both mixed blocks to vanish, giving the displayed form. Thus after a vacuum expectation value $h$, the physical fermion mass matrix has singular values $|ha|,|ha|,|hb|$: unbroken $S_3$ enforces a degenerate doublet. In a common nonorthonormal basis with invariant Gram matrix $K$, the doublet coefficient is $aK$; canonical normalization recovers $a{I}_2$.