(At least, I think it's simple.)(adsbygoogle = window.adsbygoogle || []).push({});

Disclaimer: I'm approaching this subject from the vantage point of a chemist, so be careful with how much lingo/jargon/rigor you lay on me

The claim is that if you have two representations of a group, [itex]\Gamma_1[/itex] and [itex]\Gamma_2[/itex], with bases [itex]\{ f_i \}[/itex] and [itex]\{ g_j \}[/itex], then the set of products [itex]\{f_i g_j\}[/itex] is a basis for the direct product representation. Fine. But in the process of showing this, the following assertion is made: If [itex]\hat R[/itex] is an element of the group, then

[tex]

\hat R(f_i g_j) = \hat R(f_i) \hat R(g_j).

[/tex]

...huh? How does that work? How do I know that if I apply a group operation to a product of basis elements, then it's just the product of the operation applied to the individual basis elements? I missed something there, because (a priori) that move is just as suspect as saying that [itex]x(yz) = (xy)(xz) = x^2yz[/itex] for [itex]x,y,z\in \mathbb R[/itex].

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# Simple Q about direct product representation of a group

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