What is Representation: Definition and 764 Discussions

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. Furthermore, the vector space on which a group (for example) is represented can be infinite-dimensional, and by allowing it to be, for instance, a Hilbert space, methods of analysis can be applied to the theory of groups. Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system.Representation theory is pervasive across fields of mathematics for two reasons. First, the applications of representation theory are diverse: in addition to its impact on algebra, representation theory:

illuminates and generalizes Fourier analysis via harmonic analysis,
is connected to geometry via invariant theory and the Erlangen program,
has an impact in number theory via automorphic forms and the Langlands program.Second, there are diverse approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology.The success of representation theory has led to numerous generalizations. One of the most general is in category theory. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. This description points to two obvious generalizations: first, the algebraic objects can be replaced by more general categories; second, the target category of vector spaces can be replaced by other well-understood categories.

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  1. P

    Understanding the D^{l}(\theta) Representation of 3D Rotations

    I'm having difficulty with the D^{l}(\theta) representation of 3D rotations what do the mean i suppose one you construct it for l = 1 you get the general rotation Euler matrix for 3-d Space, but what do the l = other integers or half integers mean physically? is the D matrices the...
  2. I

    (1/2,1/2) representation and photon

    I'm considering that, the (1/2,1/2) representation of Lorentz group is the four-vector representation, and this four-vector representation has spin 0,1. As we know, this four-vector representation can serve for the gauge field A_\mu, i.e. photon. However, the photon has spin 1. So we face a...
  3. R

    Matrix Representation of Tensors?

    How would you represent tensors as matrices? I've searched all over, and my book on GR (Wald) only has one example where he makes a matrice from a tensor, and I still don't understand the process.
  4. R

    What makes positive linear functionals always finite?

    I'm strugging with a portion of Rudin's proof. Quick statement of the bulk of the theorem: Let X be a locally compact Hausdorff space. Let A be a positive linear functional on Cc(X) (continous functions with compact support). Then (among other things), there exists a measure u() that...
  5. somasimple

    Virtual colors for atom representation?

    Hi there, I suppose there is a standard palette for atoms representation? i.e. white for hydrogen, red for oxygen? Is there a site or a link that provides such information?
  6. A

    Majorana representation of Gamma matrices.

    It is well known that at times we do need explicit representations for the Dirac gamma matrices while doing calculations with fermions. Recently I found two different expressions for Majorana representation for the gamma matrices. In one paper, the form used is: \gamma_{0} = \left(...
  7. P

    What is the Gelfand representation theory for Banach algebras?

    Hi, I'm reading about Gelfand's representation theory for Banach algebras and I have a small problem with one of the proofs. Let A be a commutative Banach algebra with unit, define \Gamma_A=\left\{\varphi:A\to\mathbb{C}:\varphi\neq0, \varphi \text{ an algebra homomorphism }\right\} to be...
  8. T

    Under what conditions does a function have a power series representation?

    Under what conditions does a function have a power series representation? I am looking for a theorem that says if a function satisfies these conditions then it has a power series representation. Or does all functions have a power series representation?
  9. M

    Strange representation of Heaviside and Delta function

    in the .pdf article http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6027/1/jfs080104.pdf i have found the strange representation \delta (x) = -\frac{1}{2i \pi} [z^{-1}]_{z=x} and a similar formula for Heaviside function replacing 1/z by log(-z) , what is the meaning ...
  10. H

    Exploring Representation Theory in Non-Complex Fields

    I'm done a basic course on representation theory and character theory of finite groups, mainly over a complex field. When the order of the group divides the characteristic of the field clearly things are very different. What I'd like to learn about is what happens when the field is not...
  11. F

    Matrix representation for a transformation

    Homework Statement Consider the linear transformation T: R2-->R2, where T(x1, x2, x3)= (3x2-x3, x1+4x2+x3) a. Find a matrix which implements this mapping. b. Is this transformation one-to-one? Is it onto? Explain. Homework Equations [T(x)]_B = ([T]_B) (x_B) The Attempt at a...
  12. B

    Spatial representation of field commutator

    Hi all! I worked for hours on this simple commutator of real scalar fields in qft: \left[\Phi\left(x\right),\Phi\left(y\right) \right] = i\Delta\left( x-y \right) where \Delta\left(x\right) = \frac{1}{i}\int {\frac{{d^4 p}} {{\left( {2\pi } \right)^3 }}\delta \left( {p^2 - m^2 }...
  13. N

    Irreducible representation of C3v

    Hello to everyone I'm a newcomer to the blog. I'm studying group theory and I'm dealing with irreducible representations (irreps) of C3v. Now since C3v is invariant after 6 symmetry operations I expect, as a consequence of the well known relationship g=Sum_i n^2_i (where g is the order of the...
  14. L

    Is this an accurate representation?

    Is this an accurate representation of the dimensions of string theory? http://youtube.com/watch?v=qU1fixMAObI
  15. J

    Representation of j=1 rotation matrix

    [SOLVED] Representation of j=1 rotation matrix The derivation of this involves the use of the following fact for j=1: [atex]\frac{J_y}{\hbar} = (J_y/\hbar)^3[/itex]. Is there a simple way to see this other than slogging through the algebra by expanding out the RHS using J_y =...
  16. M

    Momentum to Energy Representation in 3D Box

    Consider a 3D particle in a square box. One can represent a complete set of quantum states by indexing them with their momentum component quantum numbers. So a state would be |p_x p_y p_z>. If one goes to the energy basis, two momentum states (say |1 0 0> and |0 1 0>) will correspond to the same...
  17. F

    Why Does Differentiating a Geometric Series Lead to an Alternating Series?

    Homework Statement Use differentiation to find a power series representation for f(x) = 1/ (1+x)^2Homework Equations geometric series sum = 1/(1+x) The Attempt at a Solution (1) I see that the function they gave is the derivative of 1/(1+x). (2) Therefore, (-1)*(d/dx)summation(x^n) =...
  18. M

    Potencial as a representation of S3

    Homework Statement I am asked to write the most general real scalar potencial (without SU(2)xU(1) structure and without spin) with a irreducible representation of the symmetric group S_3. I am suppose to write it with: i) one singlet and one doublet of S_3 ii)two doublets of S_3...
  19. A

    What is the permutation representation of a cyclic group of order n?

    Homework Statement a. Find the permutation representation of a cyclic group of order n. b. Let G be the group S3. Find the permutation representation of S3. Homework Equations n/a The Attempt at a Solution I unfortunately have not been able to come up with a solution. I really...
  20. V

    Why Choose Liouville Representation Over Hamiltonian in Classical Mechanics?

    Hi All, I am interested what are the advantages, in general, of using Liouville representation instead of Hamiltonian representation and for what kind of problems such advantages are valid?
  21. P

    Understanding Coset Representation and its Role in Group Theory

    Is there a thing called a coset representation? If so is it the element that factors outside the coset? i.e. {4,8,12,16,...} = 4{1,2,3,4,...} so 4 is the coset representation for {4,8,12,16,...}
  22. R

    Find power series representation

    Homework Statement Find a power series representation for the function and determine the radius of convergence. heres the problem: http://img301.imageshack.us/img301/4514/30437250jj2.png Homework Equations The Attempt at a Solution i believe the derivative of arctant =...
  23. D

    Phasor representation of plane wave propagation

    Homework Statement I was looking through my notes when I saw the following expression of a plane wave represented as a phasor A_{0}e^{i(\vec{k}\cdot\vec{r}-\omega t)} Now I can certainly understand a plane wave propagating along a given coordinate axis say, x or z, and the phasor...
  24. S

    Polar Representation of Sinusoidal Functions

    Homework Statement What would the geometric explanation of exp(-i*x) be? Homework Equations Exp(-i*x), i being (-1)^1/2 The Attempt at a Solution I'm pretty sure this is just a circle, created clockwise? Just want to check.
  25. V

    Find the phasor representation of an equation

    Homework Statement Find the phasors of the following time functions: (a) v(t)\,=\,3\,cos\left(\omega\,t\,-\,\frac{\pi}{3}\right) (b) v(t)\,=\,12\,sin\left(\omega\,t\,+\,\frac{\pi}{4}\right) (c) i(x,\,t)\,=\,2\,e^{-3\,x}\,sin\left(\omega\,t\,+\,\frac{\pi}{6}\right) (d)...
  26. Jim Kata

    How Can I Understand the Representation Theory of Lie Algebras?

    I'm trying to derive the Gell Mann matrices for fun, but I really don't know representation theory. Can somebody help me? I have Hall's and Humphrey's books, but only online versions and staring at them makes me go crosseyed, and I'm to broke to buy a real book on representation theory. Let...
  27. F

    Set theory representation of material implication

    Just checking here. Propositional logic connectives like AND and OR have analogs or representations in set theory. For example, the logical connective AND is represented in set theory by intersection, an element of X AND Y is the element of the intersection of sets X and Y. And similarly, the...
  28. E

    Representation with Fibonacci numbers

    [SOLVED] Representation with Fibonacci numbers Homework Statement Let k >> m mean that m \geq m+2. Show that every positive integers n has a unique representation of the form n = F_{k_1} + F_{k_2} + \cdots F_{k_r}, where F_{k_i} are Fibonacci nuimbers and k_1 >> k_2 >> \cdots k_r >> 0...
  29. P

    Combinatorial group and representation theory?

    Why is there 'combinatorial' in front of both of combinatorial group theory and combinatorial representation theory? What does it imply? What is the difference and similiarities between cgt and crt besides the fact that they both have the word combinatorial in front of them?
  30. P

    Lie Groups and Representation theory?

    What is the connection between the two if any? What kind of algebra would Lie groups be best labeled under?
  31. C

    An irreducible representation of

    "An irreducible representation of..." So I was reading this paper by Max Tegmark linked from another thread, and one particular thing he said-- although didn't really have anything to do specifically with the paper it was part of-- caught my eye: So this is something I've actually seen...
  32. rocomath

    Power series representation, I really :-]

    [SOLVED] Power series representation, I really need help! :-] please let me know if i did this correctly f(x)=\arctan{(\frac{x}{3})} f'(x)=\frac{\frac{1}{3}}{1-(-\frac{x^2}{3^2})} \frac{1}{3}\int[\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{3^{2n}}]dx...
  33. L

    Spin-1/2 systems, Spinor representation

    I'm totally confused on the relationship between kets written | \uparrow \rangle, | \downarrow \rangle and | \uparrow \uparrow \rangle, | \uparrow \downarrow \rangle | \downarrow \uparrow \rangle, | \downarrow \downarrow \rangle (Problem) I have a system of two spin-1/2 particles...
  34. A

    How does a scalar transform under adjoint representation of SU(3)?

    I read this in a paper: Suppose there is a theory describing fermions transforming nontrivially under SU(3) gauge symmetry. L = \Psi^{\bar}(\gamma^A D_A+Y(\Phi))\Psi. The covariant derivative is: D_A\Psi=(\partial_A-i E_A^{\alpha}T_{\alpha})\Psi. Where E_A^{\alpha} are SU(3) gauge fields...
  35. pellman

    How to work in the momentum representation?

    In plain old QM, we can in principle take any state and expand it either in terms of momentum eigenstates or position eigentstates (i.e., the wave function). But in practice this usually means solving the Schrodinger equation, i.e., working in the postion representation from the start and then...
  36. A

    Series convergence representation

    Homework Statement \sum_{n=0}^\infty (0.5)^n * e^{-jn} converges into \frac{1}{1-0.5e^{-jn}} Prove the convergence. Homework Equations Power series, and perhaps taylor & Macclaurin representation of series. The Attempt at a Solution This isn't a homework problem...
  37. Astronuc

    Phasor representation of AC voltage and current

    Phasor representation of AC voltage and current. I\,=\,5\angle{0^o}\,=\,5\,+\,j0\,A V\,=\,100\angle{30^o}\,=\,86.6\,+\,j50\,V in general V\,=\,A\angle{\theta^o}\,=\,A cos{\theta}\,+\,jA sin{\theta}\,V and similarly for I It is assumed that the angular frequency \omega is...
  38. S

    Parametric Representation of a Helix

    Just wanted to check and see if this is right. The k-component of the vector is what I'm unsure of...I've always sucked at converting to parametric form. :) Homework Statement Convert to parametric form: x^{2} + y^{2} = 9, z = 4arctan(y/x) The Attempt at a Solution The i- and...
  39. B

    Power series representation of 10xarctan(5x).

    Homework Statement The function f(x)=10xarctan(5x) is represented as a power series http://img464.imageshack.us/img464/4131/formub5.jpg Find the first few(5) coefficients in the power series. Homework Equations I already know that the representation of arctanx is summation from...
  40. Q

    Matrix representation of complex space

    more on the complexification of a vector space. if we complexify some space V, then how does the matrix representation of some linear T on V relate to the matrix representation of T on Vc on the standard ordered basis for V?
  41. I

    Matrix Representation of S_z Using Eigenkets of S_y

    Show the matrix representation of S_z using the eigenkets of S_y as base vectors. I'm not quite sure on the entire process but here's what i think: We get the transformation matrix though: U = \sum_k |b^{(k)} \rangle \langle a^{(k)} | where |b> is the eigenket for S_y and <a| is...
  42. A

    Parametric Representation of a Plane

    Homework Statement Give a parametric representation of the plane x + y + z = 5. Homework Equations I am really not sure, I've been over the chapters we've covered for a little over an hour now, and the only mention i can find of a parametric representation of a plane is in passing...
  43. I

    Matrix Representation: What Happens to |a''><a'|?

    let X be an operator, we can write X as a matrix where: X = \sum_{a''} \sum_{a'} |a''><a''|X|a'><a'| where <a''|X|a'> a'' are the rows and a' are the columns. I was wondering what happened to the |a''> <a'|? It seems like they are disregarded when transforming to matrix notation. I...
  44. P

    Exploring Representation Theory: History, Difficulty, and Connections to Physics

    I am going to ask some general questions about representation theory which may sound stupid as I don't know anything about it. How old or new is this theory? How did it come out? What is the general consensus of ths difficulty of the subject? Are there many open problems in this theory? Is...
  45. H

    Young Tableux and representation theory

    Hey guys, I was just wondering if you have a good references to the use of Young diagrams/Tableuxs specifically to deduce the representation theory of various Lie groups *other than SU(N)* I know how it works for SU(N) and those groups that can be split into tensor copies theoreof, but I have...
  46. R

    Why Adjoint Representation Has a Higher Dimension Than Basis Matrices?

    Why has the adjoint representation a higher dimension than the basis matrices it acts on? for example here Why is e_1 two dim and ad(e_1) four dim? Isn't ad(X) Y a simple matrix multiplication here? But then multiplying 4x4 with 2x2 matrices, what does it mean? thanks
  47. D

    When is the Interaction representation used and why?

    Hi, Can someone explain to me why do we actually need the Interaction/Intermediate representation? In my past, each course in QM touched it only for a few minutes and then it got... forgotten. Can someone please give me an example as to how (and when) it is used (and a good reason why)...
  48. J

    Path integral in momentum representation

    Is it possible to derive the Shrodinger's equation i\hbar\partial_t \Psi(t,p) = \frac{|p|^2}{2m}\Psi(t,p) in momentum representation directly from a path integral? If I first fix two points x_1 and x_2 in spatial space, solve the action for a particle to propagate between these...
  49. T

    Complex representation of fourier series

    Homework Statement Using the complex representation of Fourier series, find the Fourier coefficients of the periodic function shown below. Hence, sketch the magnitude and phase spectra for the first five terms of the series, indicating clearly the spectral lines and their magnitudes...
  50. kakarukeys

    Sufficient Condition for Adjoint Representation Kernel to be Lie Group Center

    what is the sufficient condition for the kernel of an adjoint representation to be the center of the Lie group? Does the Lie group have to be non-compact and connected, etc?
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