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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...

  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.

    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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