= Invariant bilinear forms of odd dihedral doublets
{title2=$D_k\otimes D_k\supset\mathbf1$}
For the <dihedral group> of order $2m$ with odd $m$, let $D_k(r)=\operatorname{diag}(\zeta^k,\zeta^{-k})$ and $D_k(s)=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, where $\zeta=e^{2\pi i/m}$ and $1\leq k\leq(m-1)/2$. An <invariant tensor> defining a bilinear form satisfies $D_k(g)^TLD_k(g)=L$. The rotation forces $L_{11}=L_{22}=0$ and the reflection forces $L_{12}=L_{21}$. Thus the <invariant tensor> is a multiple of $\begin{pmatrix}0&1\\1&0\end{pmatrix}$, and a singlet <Yukawa interaction> gives two equal physical masses by <singular value decomposition>. Distinct inequivalent doublets admit no such bilinear form, by <Schur lemma> and their self-duality.
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