Question about reducible presentation

In summary, the conversation discusses finding the reducible representation for a base consisting of 8 unit vectors for a planar four-atomic molecule. It also mentions decomposing the representation into irreducible representations and determining the number of vibrational modes in the xy-plane. The speaker is unsure about how to determine the reducible representation and suggests that it may be a 8x8 matrix. They also mention that there should be 6 such matrices due to the symmetry of BF3 being D3h.
  • #1
lanbeiming
7
0
1.hi, everyone. here's the question :BF3 is a planar four-atomic molecule. For simplicity, we ignore the degrees of
freedom in the z-axis (principal axis).
a) Find the reducible representation for a base consisting of the 8 unit vectors with origin
at each atom, being parallel to the x and y directions.
b) Decompose it into a sum of irreducible representations. How many vibrational modes
in the xy-plane exist? i know how to get the irreducible linear representation from the reducible representation, but i hardly gain any idea about how to determine the reducible representation which help me for the later calculation to the irreducible representation.

i guess wether the reducible representation is a 8*8 matrix corresponding to the each (x.y) coordinates for the four atoms. then there should have 6 such matrix since BF3 symmetry is D3h. But i don't know then how to gain the irreducible representation from such matrix
 
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  • #2
where's your attempt at solution?
 

What is a reducible presentation?

A reducible presentation is a mathematical concept used in group theory. It is a way of representing a group as a combination of simpler groups, known as factors, through multiplication.

How is a reducible presentation different from a normal presentation of a group?

In a normal presentation, a group is represented by generators and relations. In a reducible presentation, the group is represented by factors and a multiplication operation. This allows for a more structured and organized representation of the group.

How is a reducible presentation useful in studying groups?

A reducible presentation allows for a deeper understanding of the structure and properties of a group. By breaking down a group into simpler factors, we can analyze each factor individually and then combine them to understand the overall group.

Can any group be represented by a reducible presentation?

No, not every group can be represented by a reducible presentation. This type of representation is only applicable to certain types of groups, such as finite groups or certain types of infinite groups.

Can a group have multiple reducible presentations?

Yes, a group can have multiple reducible presentations. Just like how a group can have multiple normal presentations, it can also have different reducible presentations depending on how it is broken down into factors.

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