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Product Groups and their dimensions

  1. Jul 23, 2012 #1
    My understanding was that the product of two groups A and B will yield a group C for which the dimension of C is dim(A)*dim(B).
    Now however, the author I'm reading defines the group product multiplication as:

    (a1, b1) * (a2, b2) = (a1*a2, b1*b2), for a1,a2 in A and b1, b2 in B.

    Does this give the same result for the dimension??



    Lets take the example with A and B corresponding to the real numbers with addition as a group operation. This gives:

    (x1, y1) + (x2, y2) = (x1+x2, y1+y2).... It looks like we have created the two dimensional plane! But dim(R)*dim(R) = 1 * 1 = 1. C should have dimension 1 not 2. :/


    Please help me!
     
  2. jcsd
  3. Jul 23, 2012 #2

    What do you mean by "dimension of a group"? I think there's a huge confusion here...

    DonAntonio
     
  4. Jul 23, 2012 #3
    By the dimension of the group I mean the parameters we need to describe the group.. For the real numbers and U(1) this is 1. For SU(3) this is 3, etc.
     
  5. Jul 23, 2012 #4
    I think the "dimension" of the direct product of two groups is the sum of the corresponding dimensions.

    This is because of the exponentiation of Lie groups.

    Edit:
    After reviewing what you wrote, I am puzzled as to how you found the dimension of SU(3) to be 3?
     
  6. Jul 23, 2012 #5
    Gah! hahah, confusing.
    I think Im confusing the tensor product which I have encountered in the representations theory of groups and the group product.. For the tensor product the dimension of the representations is clearly multiplicative.. then I assumed that the same would be true for the group product.



    It should of course be dimension 2 for SU(3)- it is topologically equivalent to the 2-sphere. :)
     
  7. Jul 23, 2012 #6
    These are incorrect statements.
     
  8. Jul 23, 2012 #7
    Lets skip the SO(3) for the moment, and focus on the topic.


    What is the relation between the tensor product between two representation and the correponding product group? :/
    Can anything be said here?
     
  9. Jul 23, 2012 #8
    Yes, since representations are nothing more but matrices, and the dimension of the Kronecker product is the product of the dimensions, the dimension of the direct tensor product representation would be a product of the dimensions of the representations of each group.
     
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