= Symmetric tensor vacuum stabilizer
{title2=$H=S(\prod_iO(m_i))$}
A real <symmetric tensor> of the <special orthogonal group> transforms as $Q\mapsto OQO^T$. If its <vacuum expectation value> has distinct <eigenvalues> with multiplicities $m_1,\ldots,m_p$, its <stabilizer subgroup> is $S(O(m_1)\times\cdots\times O(m_p))$: each transformation preserves each <eigenspace> and the product of the block <determinants> is one. Its <identity component> is $\prod_iSO(m_i)$. Thus a scalar multiple of the identity preserves the whole <group>, whereas a tensor with simple spectrum preserves only diagonal signs with total sign one. A tracelessness condition constrains the eigenvalues but does not change this stabilizer formula.
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