= S3-invariant Yukawa matrix for a doublet and a sign singlet
{c}
{title2=$\lambda=\operatorname{diag}(a,a,b)$}
Let the left and right <fermions> transform in $D\oplus\varepsilon$, where $D$ is the two-dimensional <irreducible representation> of the <symmetric group> $S_3$ and $\varepsilon$ 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 $D\oplus\varepsilon$. 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 $aI_2$.
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