Group representations Definition and 28 Threads

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations (i.e. automorphisms) of vector spaces; in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations.

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  1. redtree

    I Generators of translations

    I just want to make sure I understand this correctly. For an infinite-dimensional representation, the generators of translation can be written as ##i \frac{\partial}{\partial_{\mu}}= i \partial_{\mu}##, where the generators of the Lorentz group can be written as ##i (x^{\mu}\partial_{\nu} -...
  2. K

    I Change of basis matrix for point group C3V

    I am looking at the point group C<sub>3v</sub> described shown here. I am trying to understand the block diagonalization process. The note says that changing the basis in the following way will result in the block diagonal form. What is the rationale for choosing the new basis. Is it...
  3. M

    Gauge invariance in a non-abelian theory SU(2)xU(1)xU(1)

    I believe what is asked is impossible. Here is why. The U(1) factors are abelian, so V and T commute with each other and with U, so i can just try to build a term containing and even number of T-s,V-s and U-s. From the transformation laws we see that a bilinear term in the Weyl fermions must...
  4. H

    B Tensor product of operators and ladder operators

    Hi Pfs i have 2 matrix representations of SU(2) . each of them uses a up> and down basis (d> and u> If i take their tensor product i will get 4*4 matrices with this basis: d>d>,d>u>,u>d>,u>u> these representation is the sum equal to the sum of the 0-representation , a singlet represertation with...
  5. Rabindranath

    I Meaning of terms in a direct sum decomposition of an algebra

    Let's say I want to study subalgebras of the indefinite orthogonal algebra ##\mathfrak{o}(m,n)## (corresponding to the group ##O(m,n)##, with ##m## and ##n## being some positive integers), and am told that it can be decomposed into the direct sum $$\mathfrak{o}(m,n) = \mathfrak{o}(m-x,n-x)...
  6. pellis

    I Calculating group representation matrices from basis vector/function

    Being myself a chemist, rather than a physicist or mathematician (and after consulting numerous sources which appear to me to skip over the detail): 1) It’s not clear to me how one can go generally from a choice of basis vectors in real space to a representation matrix for a spatial symmetry...
  7. lelouch_v1

    I Quark Model Families and Masses

    Consider the pseudoscalar and vector meson family, as well as the baryon J = 1/2 family and baryon J = 3/2 family. Within each multiplet, for each particle state write down its complete set of quantum numbers, its mass, and its quark state content. Furthermore, for each multiplet draw the (Y...
  8. Ramtin123

    A Why Is the Mixed SU(2) Term Invariant in Scalar Multiplet Models?

    Consider two arbitrary scalar multiplets ##\Phi## and ##\Psi## invariant under ##SU(2)\times U(1)##. When writing the potential for this model, in addition to the usual terms like ##\Phi^\dagger \Phi + (\Phi^\dagger \Phi)^2##, I often see in the literature, less usual terms like: $$\Phi^\dagger...
  9. tomdodd4598

    I Exploring Direct Sums of Lorentz Group Representations

    Hey there, I've suddenly found myself trying to learn about the Lorentz group and its representations, or really the representations of its double-cover. I have now got to the stage where the 'complexified' Lie algebra is being explored, linear combinations of the generators of the rotations...
  10. A

    I Adjoint Representation Confusion

    I'm having a bit of an issue wrapping my head around the adjoint representation in group theory. I thought I understood the principle but I've got a practice problem which I can't even really begin to attempt. The question is this: My understanding of this question is that, given a...
  11. S

    I What are the best resources for learning about Lorentz group representations?

    Hello! Can someone recommend me some good reading about the Lorentz group and its representations? I want something to go pretty much in all the details (not necessary proofs for all the statements, but most of the properties of the group to be presented). Thank you!
  12. J

    A How is the invariant speed of light enocded in SL(2,C)?

    In quantum field theory, we use the universal cover of the Lorentz group SL(2,C) instead of SO(3,1). (The reason for this is, of course, that representations of SO(3,1) aren't able to describe spin 1/2 particles.) How is the invariant speed of light enocded in SL(2,C)? This curious fact of...
  13. Ravi Mohan

    Poincare group representations

    My question concerns both quantum theory and relativity. But since I came up with this while studying QFT from Weinberg, I post my question in this sub-forum. As I gather, we first work out the representation of Poincare group (say ##\mathscr{P}##) in ##\mathbb{R}^4## by demanding the Minkowski...
  14. Anchovy

    Standard Model decompositions of larger group representations?

    When reading about GUTs you often come across the 'Standard Model decomposition' of the representations of a given gauge group. ie. you get the Standard Model gauge quantum numbers arranged between some brackets. For example, here are a few SM decompositions of the SU(5) representations...
  15. B

    Algebra Good book on representation theory of groups

    Hi I am a physics graduate student. Recently I am learning representation theory of groups. I understand the basic concepts. But I need a good book with lots of examples in it and also exercise problems on representation theory so that I can brush up my knowledge.The text we follow is "Lie...
  16. Dilatino

    How can I construct the 4D real representation of SU(2)?

    An element of SU(2), such as for example the rotation around the x-axis generated by the first Pauli matrice can be written as U(x) = e^{ixT_1} = \left( \begin{array}{cc} \cos\frac{x}{2} & i\sin\frac{x}{2} \\ i\sin\frac{x}{2} & \cos\frac{x}{2} \\ \end{array} \right) = \left(...
  17. S

    Meaning of representations of groups in different dimensions

    Problem This is a conceptual problem from my self-study. I'm trying to learn the basics of group theory but this business of representations is a problem. I want to know how to interpret representations of a group in different dimensions. Relevant Example Take SO(3) for example; it's the...
  18. I

    Scalar as one dimensional representation of SO(3)

    Hi to all the readers of the forum. I cannot figure out the following thing. I know that a representation of a group G on a vector spaceV s a homomorphism from G to GL(V). I know that a scalar (in Galileian Physics) is something that is invariant under rotation. How can I reconcile this...
  19. ShayanJ

    Some questions about group representations

    I was rethinking about some things I learned but I came to things that seemed to be not firm enough in my mind. 1) When we want to find the unitary matrix that block-diagonalizes a certain matrix through a similarity transformation, we should find the eigenvectors of that matrix and stick them...
  20. S

    Are Vectors Defined by Commutation Relations Always Roots in Any Representation?

    The vectors \vec{\alpha}=\{\alpha_1,\ldots\alpha_m \} are defined by [H_i,E_\alpha]=\alpha_i E_\alpha they are also known to be the non-zero weights, called the roots, in the adjoint representation. My question is - is this connection (that the vectors \vec{\alpha} defined by the commutation...
  21. M

    Equality involving matrix exponentials / Lie group representations

    We have that A and B belong to different representations of the same Lie Group. The representations have the same dimension. X and Y are elements of the respective Lie algebra representations. A = e^{tX} B = e^{tY} We want to show, for a specific matrix M B^{-1} M B = AM Does it suffice to...
  22. V

    Spin orbit and double group representations

    I am reading a text about the splitting of the energy levels in crystals caused by the spin orbit interaction. In particular, the argument is treated from the point of view of the group theory. The text starts saying that a representation (TxD) for the double group can be obtained from the...
  23. M

    Functional Analysis or group representations?

    I have to choose a total of 12 modules for my 3rd year. I've everything decided except four of them. I want to eventually do research either General Relativity, quantum mechanics, string theory, something like that. I'm torn between Group Representations, with one of Practical numerical...
  24. ArcanaNoir

    Group representations, interesting aspects?

    I am writing an undergraduate "thesis" on group representations (no original work, basically a glorified research paper). I was wondering if anyone could suggest interesting aspects that might be worth writing about in my paper. I have only just begun to explore the topic, and I see that it...
  25. R

    Group representations on tensor basis.

    I am a physicist, so my apologies if haven't framed the question in the proper mathematical sense. Matrices are used as group representations. Matrices act on vectors. So in physics we use matrices to transform vectors and also to denote the symmetries of the vector space. v_i = Sum M_ij...
  26. W

    Characteristic classes of finite group representations

    I know zero about the characteristic classes of finite group representations and would appreciate a reference. specifically, if I have a faithful representations of a finite group,G, in O(n) what can I say about the induced map on cohomology, P*:H*(BO(n))-> H*(BG) ? I am mostly interested in...
  27. J

    Understanding Group Representations in Group Theory

    Hey guys, I'm pretty new to group theory at the moment, what's the best way of understanding a 'representation' of a group? Thanks
  28. A

    Group Representations and Young Tableaux

    What are good resources on Young diagrams and tableaux for representations of the permutation groups Sn and the unitary groups U(n) of n x n unitary matrices?
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