# 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?

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.