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In regards to direct product representations, I'm trying to find a proof for:

$R(X_iY_j)=R(X_i)R(Y_j)$

Where $R$ is a symmetry operation of a group and $X_i$ and $Y_j$ are members of different basis sets for the group. See e.g. Chemical Applications of Group Theory by F.A. Cotton, 3rd ed., p.105 or Atkins' Molecular Quantum Mechanics, 5th ed.,p.152.

A lot of books I've seen covering direct product representaions start with this equality but I haven't seen any justification for it yet.

Any help is greatly appreciated!

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  • $\begingroup$ You could try D. Bishop 'Group Theory & Chemistry' publ Dover 1973, chapter 8 p 156 where he expands the product of operators as two matrices. (its not easy) $\endgroup$
    – porphyrin
    Commented Oct 16, 2020 at 8:01

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