# Given a VEV how can I compute which generators remain unbroken using tensor methods?

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Let's assume we have Higgs fields in some representation $R$, of a gauge group $G$ as for example $SO(10)$, that can be written as a matrix. Then given a vev written as a matrix how can I compute which generators remain unbroken after the Higgs fields get a vev?

For concreteness, let's consider the $SO(10)$ example. The gauge bosons live in the adjoint 45-dim rep. Because $10 \otimes 10= (1 \oplus 54)_S \oplus 45$, we can write each gauge boson as a (antisymmetric) $10 \times 10$ matrix $M_{45}$. Further let's assume we have Higgs fields in the $54$ dimensional representation. For the same reason we can write these 54 Higgs fields as (symmetric) $10 \times 10$ matrices $M_{54}$.

Then how can I compute which generators remain unbroken when some of the Higgs fields (or just one) gets a vev? (The vev is then a $10 \times 10$ matrices $M_{54VEV}$)

Three possibilities that come to my mind:

• $M_{45}M_{54VEV}=0$
• $M_{45}M_{54VEV}M_{45}^T=0$
• $[M_{45},M_{54VEV}]=0$
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