# Difference between A and B irreps

• I
• ftft
In summary, in the conversation it is discussed that in group theory, the one-dimensional irreps A and B differ in their symmetry under rotation about the principal axis. B is when the state is antisymmetric while A is symmetric. However, in the character table of point group D, B1 has a character of 1 under the principal rotation while B2 and B3 have a character of -1. It is questioned why B1 is still referred to as B1 even though it has a character of 1 when rotated about the principal axis. The reason for this is because in this particular group, the three axes are equivalent, so the choice of the principal rotation axis is arbitrary.

#### ftft

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?

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.

ngonyama said:
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.

## 1. What is the difference between A and B irreps?

The difference between A and B irreps lies in their mathematical properties and their role in the representation theory of groups. A and B are typically used to represent different types of symmetry operations, with A being used for operations that preserve orientation, and B being used for operations that reverse orientation.

## 2. How are A and B irreps related?

A and B irreps are related through their connection to the symmetry operations of a group. They are both part of a larger set of irreps that represent the various symmetries of a system. The relationship between A and B irreps can also be seen in the way they transform under the group's operations.

## 3. Can A and B irreps be combined?

Yes, A and B irreps can be combined to form a new representation. This new representation will have different mathematical properties and will represent a different type of symmetry operation. The combination of A and B irreps can also result in a reducible representation, where it can be broken down into smaller, irreducible representations.

## 4. What is the physical significance of A and B irreps?

The physical significance of A and B irreps lies in their connection to the symmetries of a system. These symmetries can have a profound impact on the physical properties and behavior of a system. By studying the irreps, scientists can gain a better understanding of the symmetries and how they influence the system.

## 5. How are A and B irreps used in physics?

A and B irreps are used in physics to study the properties of a system and to understand the underlying symmetries. They are also used in the development of mathematical models and theories to explain physical phenomena. In addition, A and B irreps are frequently used in quantum mechanics and particle physics to describe the symmetries of particles and their interactions.