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In character tables, I know that A and B are one-dimensional irreducible representations, but what is the difference them?

And why do some character tables list them as A1/A2, B1/B2 and others have Au/Ag, Bu/Bg?

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    $\begingroup$ g|u are from the German terms for even|odd, 1|2 are for people who doesn't care to remember those, and the difference is all there in the tables. Also, welcome to Chem.SE. $\endgroup$ Aug 5, 2016 at 7:02

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The traditional labelling of irreducible representations is the following:

  • A, when rotation around the principal axis is symmetrical
  • B, when rotation around the principal axis is asymmetrical

You can see, for example, that $C_n$ groups of even order have both A and B representations, while those of odd order cannot feature B representations.

As others have commented, there are additional rules… E for doubly degenerate representations, T for triply degenerate, u and g for parity under inversion (odd and even, respectively), greek capital letters Σ, Π, Δ for infinite groups (C∞v and D∞h).

Finally, if there are several irreducible representations that are not distinguished under those conventions, they are numbered.

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  • $\begingroup$ @ F'x, its a small point but it would it not be better to say for A, that after rotation about the principle axis the molecule is indistinguishable compared to that before rotation, (or words to this effect) and in B it is distinguishable. I suggest this because symmetrical/asymmetrical are either not exact enough or confusing in this context. $\endgroup$
    – porphyrin
    Aug 5, 2016 at 13:35

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