# Difference between A and B irreps

• I
In my understanding, in group theory the one-dimensional irrep A differs from the one-dimensional irrep B in the symmetry under rotation about the principal axis: A is when the state is symmetric and B is when the state is antisymmetric under that rotation. However, I find in the character table of point group D that B1 has a character of 1 under the principal rotation operation while B2 and B3 have a character of -1.
Why the irrep is called B1 even though it has a character of 1 when rotated about the principal axis?

## Answers and Replies

jim mcnamara
Mentor
You will get improved answers if you can tell us what you are reading, exactly e.g., Foobar and Flatwheel, Chap 5, p. 96 ( This is meant to be an example only)

Do you mean the group D_2?

There are three B irreps there. I think because it is not clear what the principal rotation axis is: the three axes are equivalent.

Do you mean the group D_2?

There are three B irreps there. I think because it is not clear what the principal rotation axis is: the three axes are equivalent.
Thanks for the reply.. Yes, I meant D_2 group.. But why are the 3 axes equivalent? I thought the rule is that one principal axis in the system is chosen to define the properties of the irreps. Say we choose the z-axis to define the principal axis, B1 behaves like A2

Well, they are all two-fold axes. So, which one you choose to be the unique z-axis is really up to you. In groups like D_3 D_4 that is not so because one is a three- or fourfold and perpendicular to that you have twofolds. So then you should pick the higher order axis to be the main z-axis.