= Antisymmetric tensor vacuum stabilizer in SU(4)
{title2=$A=\operatorname{diag}(a\epsilon,b\epsilon)$}
For a complex <antisymmetric tensor> $A\in\wedge^2\mathbb C^4$, the <SU(4)> action is $A\mapsto UAU^T$. The <unitary skew-diagonalization of an antisymmetric matrix> gives $A=\operatorname{diag}(a\epsilon,b\epsilon)$ in suitable coordinates, where $\epsilon=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$. If $|a|\ne|b|$, preserving $AA^\dagger$ forces a stabilizing <unitary matrix> to preserve the two coordinate planes. The identity $V\epsilon V^T=(\det V)\epsilon$ then forces both two-by-two blocks to have <determinant> one, using $\det U=1$ also when one of $a,b$ vanishes. The exact <stabilizer> is $SU(2)\times SU(2)$. If $|a|=|b|\ne0$, the <stabilizer> is the larger <compact symplectic group> $USp(4)$. If $a=b=0$, it is all of $SU(4)$. The <real structure of the SU4 exterior square> imposes $b=\bar a$ in this form, so a nonzero real six-dimensional field has the larger symplectic <stabilizer>, whereas unequal paired <singular values> are generic for a complex six-dimensional field.
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